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CHAPTER 2

REVIEW OF LITERATURE

To better understand mathematics in today's world, Chapter 2 will examine concepts in mathematics. A historical perspective of mathematics curricula will be explored in addition to the reform movement, traditional mathematics, real-world applications, and integrated mathematics.

Numerous factors have influenced the mathematics curriculum in the United States. Of particular importance are: (a) societal forces, (b) how children learn mathematics, and (c) the nature of mathematics subject matter (NCTM, 1989). These three factors have interacted with research in psychological learning theories to influence the mathematics curriculum (Kroll, 1989). Factors influencing real-world mathematical problem-solving include: (a) psychological learning theories, (b) traditional mathematics, (c) connections in mathematics, (d) the integration of mathematics, and (e) mathematical approaches.

A mathematics curriculum developed and taught by Sumerian scribes in 3,000 B.C. was thought by Kilpatrick (1992) to be the basis for mathematics education in the United States today. Using applied mathematics, the Sumerian scribes developed means of teaching place value, sexagesimal fractions, and the use of tables for calculation. In the 5th century B.C., Plato's Meno related an incident of Socrates using skillful questioning to teach the concept that the area of a square on the diagonal of another square is twice that of the smaller square. Kilpatrick further revealed that 16th-century mathematics teacher Robert Recorde employed textbooks that dealt with Socratic dialogue of proof, definition, and understanding.

In addition to these influences on the American mathematics curriculum, Tanner and Tanner (1980) recounted the 1894 tenets of Pestalozzia and Froebel, educators and psychologists who proposed teaching methods based on concrete experiences and educational aims concerned with the development of mental faculties that influenced the teaching of mathematics from kindergarten through high school. According to Tanner and Tanner, Froebel's philosophy of "learning by doing" contributed to the United States' inclusion of students manipulating objects, exploring, and expressing themselves in mathematics.

Kilpatrick (1992) pointed out that Warren Colburn published in 1821 an innovative arithmetic textbook based on Pestalozzia's pedagogy, which gave students the opportunity to discover rules by induction from examples. Jahnke (1986) related the history of Martin Ohm, who published several works on mathematics education from 1816 to 1822, which advocated mathematics as a system of relations between operations. Jahnke stated that Ohm later wrote influential textbooks in which the number concept was organized as a progressive extension from natural numbers to rational, negative, real, and complex numbers.

At the end of the 19th century, observed Tanner and Tanner (1980), educational theorists once again became more interested in curriculum. Throughout the 20th century, they concluded, conflict coexisted between the old and new way of looking at curriculum. They noted that, during the first two thirds of the 20th century, mathematics teaching in American elementary and secondary schools passed through three major phases: the drill-and-practice, the meaningful, and the new mathematics.

Resnick and Ford (1981) maintained that educational theorist E. L. Thorndike, who was the "founding father" of the psychology of mathematics instruction, greatly influenced the drill-and-practice phase in the mathematics curriculum. They pointed out that his associationism, a justification for the theory of drill-and-practice that he published in 1922, was rooted in a tradition of laboratory experimentalism and that he was also strongly guided to the task of translating laboratory findings into guidelines for classroom instruction. Thorndike (1922) primarily experimented with animals, but thought his learning principles should apply to humans; his law of effect suggested that practice followed by reward was an important way in which human learning took place. He believed that what teachers needed was to find and make explicit the particular set of bonds that constituted arithmetic.

Thorndike (1922) reasoned that the task of instruction was to form carefully the necessary bonds and habits that would allow students to perform computations and solve problems. As a first step, one would have to select bonds to be formed. Any carefully constructed arithmetic curriculum would divide the subject matter into broadly defined topics. Once the proper bonds were selected, they could be formed and strengthened. This was where drill-and-practice came in.

Thorndike's (1922) proper drill and practice involved presenting bonds in a carefully programmed way so that important bonds were practiced often, and lesser bonds, less often. The teacher's job was to provide the proper amount of practice, in the proper order, on each class of problems. The teacher was to identify the bonds that made up the subject matter of interest and then put them in order, easier ones first, arranging them so that learning the easier ones would help in learning the harder ones that came later in the series. Then, simply put, students had to practice each of the kinds of bonds. Each class of bonds was to be practiced just enough so that errors could be avoided when advancing to the next harder class of bonds. And since more complex problems were conceived as strings of bonds, it was important to drill well on each of the connections that would be needed for the harder problems.

William Brownell (1928) came out in opposition to the bond theory and drill method in 1928. Brownell found that third graders in the drill program were using a variety of procedures rather than direct recall to do their addition and subtraction problems. Brownell's interpretation of this was that drill simply made them faster and better at immature procedures they had discovered for themselves, not at the direct recall that adults possess. Secondly, the drill method implied a distorted view of the goal of learning. Brownell advocated that instruction should stress concepts and relationships to ensure quantitative thinking and that students are better able to apply their knowledge in novel situations.

Resnick and Ford (1981) related the work of McConnell in 1934 and 1958, which compared a rigorous drill method to a meaningful method of instruction. McConnell found the drill method to be a more straightforward avenue for automatic and immediate responses, but on measures of transfer to untaught combinations, the meaningful approach gave significantly better results.

Resnick and Ford (1981) concluded that Brownell and Thorndike differed in their definitions of what should be learned. Thorndike advocated mathematics learning consisting of a collection of bonds, whereas Brownell believed in an integrated set of principles and patterns. The two definitions required two different methods of teaching.

Moving forward a generation, Resnick and Ford (1981) summarized the work of Swenson. Swenson (1949) found that the generalization method proved most effective in promoting learning of the material and transfer to new material, with drill-plus somewhat less effective, and straight drill least effective. Resnick and Ford reported that, despite Swenson's findings, the bond theory dominated teaching and learning during this historical period in mathematical education, and that educators and parents only began questioning the extreme emphasis on drill and practice in the 1930s and 1940s. These parents questioned whether or not their children's arithmetic learning was of practical use. Hence, the progressive era began to ensure that the skills students would master constituted more meaningful arithmetic (Resnick & Ford, 1981).

Progressive education gained its momentum as a movement in the years prior to World War II. American progressivism's ultimate goal was to improve the hidden poor's quality of life (Tanner & Tanner, 1980). Mathematics education focused on developing arithmetic concepts in a meaningful way. Thus, the activity-oriented approach to teaching mathematics began.

Another important factor in the emergence of the new meaningful theory of arithmetic was the introduction in the United States of the Gestalt theory of learning. This theory emphasized an organized whole as contrasted with a collection of parts. These Gestalt psychologists regarded learning as a process of recognizing relationships and of developing insights. Thus, the activity-oriented approach to teaching mathematics assisted students in understanding connections between the many skills and concepts (Resnick & Ford, 1981).

Gestalt psychologists in the 1920s eventually focused on a more general problem: the nature of thinking and problem-solving. One focus of problem-solving was the phenomenon of insight. Kohler (1925), who had worked with Wertheimer on early experiments on perception, observed closely the behavior of a captive colony of chimpanzees over a number of years. He particularly noted their efforts in solving everyday problems. For example, finding that a banana was high up on a shelf, a chimpanzee would turn away from the food and move to the other side of the cage to procure a box to stand on. In another example, an ape was given two sticks to play with, one of which could be fit into the other to make a single longer stick. A banana lay on the floor just out of reach outside the cage. After a period of play with the sticks, the ape suddenly seemed to understand how they might be used to solve the problem. The ape deliberately fitted the sticks together and used the long stick to bring the banana within his reach.

Kohler (1925) viewed problem situations for humans as creating tensions in a psychological field, just as the blocked goal initiated solution activity in apes. Where a solution was not immediately activated, where there was conflict in the problem situation, the dynamic mental forces would seek equilibrium in reorganization. Insight occurred at the point of this reorganization. Until the structure of the problem was understood through insight, the problem situation was not meaningful to the problem solver. Hence, the problem was not solvable.

Wertheimer (1959) was interested in demonstrating an appreciation of structure. His classic example was the parallelogram problem. Wertheimer went into a classroom where students were being taught to find the area of a parallelogram. Their teacher had shown them how to construct a line from the upper left-hand corner such that it formed a 90-degree angle with the base. They then were to measure the line and multiply it by the length of the base to find the answer. Wertheimer stepped to the front of the room and posed a problem. He showed them an upended version of a regular parallelogram and asked them to find the area. The children's reactions were mixed. The difficulty was that when the children dropped a perpendicular from the top left-hand corner, as they had been taught, the line ended up somewhere to the left of the base, so that the standard formula did not seem to apply.

Wertheimer's (1959) point was that students who learned the algorithm for finding area without understanding the structural principles were limited to following blindly the rules set forth by their teacher. They had been taught the algorithm in a rote fashion in which they did not learn it with meaning. He saw that understanding the problem structure resulted in productive thinking.

A new era in mathematics began in the 1960s. This "new math" clarified mathematical structures, and by so doing, was considered to be a refinement and extension of meaningful mathematical programs initiated during the progressive era. Value was placed on meaningful learning, but now standards for meaningfulness emerged. Mathematicians suggested that meaningful learning would benefit children if students were taught the structure of mathematics (Resnick & Ford, 1981).

Cognitive psychology introduced how mathematics might be made meaningful by responding to specific intellectual capabilities of learners. Thus, the influence of this promise of teaching mathematics' structures ensued. In addition, Jerome Bruner, a noted psychologist, supported a spiral curriculum and the idea of discovery (Resnick & Ford, 1981).

Jean Piaget (1970), an influential and noted psychologist, conducted studies relating to mathematics that continue to have instructional implications. His most extensive work was on the development of logical and mathematical concepts. He studied the growth of logical classification systems and the concepts of number, geometry, space, time, movement, and speed with children and adolescents. These topics clearly involved the use of certain logical structures. Piaget believed these structures were the basis of thinking and reasoning, particularly of a scientific kind.

Through his experiments, Piaget (1970) showed that children become increasingly more sophisticated in their thinking as they become older. This theory purports that, as people grow older, they develop new, more complex cognitive structures. The concept of an operation is central to Piaget's developmental distinctions. The presence or absence of certain operations defines the stages of development to intellectual maturity. It takes extended practice and experience for new logical structures to develop. Piaget believed that children developed through different stages from the stage of preoperational thought through the stage of concrete operations to the stage of formal operations. An instructional approach is to attempt to match instruction to children's developmental level. Instead of waiting for students to be ready for instruction, a more positive approach is to give students tasks that present a challenge which has familiar elements in it. General principles of constructive learning, concrete representations, social feedback, and clinical teacher-pupil interaction were derived from Piagetian theory (Resnick & Ford, 1981).

Constructivism is based upon the premise in mathematics education that children have a mathematical reality of their own (Steffe & Wiegel, 1992). In other words, for knowledge to be meaningful, students need to construct it themselves (Marlowe & Page, 1998). The constructivist philosophy holds that students' learning of subject matter is the product of the interaction between what they are taught and what they bring to any learning situation (Ball, 1988). Steffe and Wiegel further explained that mathematics knowledge is based on coordinations of such actions into organized patterns to achieve some goal. Students learn mathematics by actively reorganizing their own experiences in an attempt to resolve their problems (Cobb, Yackel, & Wood, 1995). Constructivism influenced all aspects of learning — the teaching, curriculum, learning, assessment, and technology.

Constructivism influenced mathematics by individuals constructing mathematical insights and their meaning within an individual's experience. Students' explanations, their inventions, have legitimate epistemological content and are the primary source of investigation. With this understanding of constructivism in mathematics, ideas in mathematics are created and their status is negotiated within a culture of mathematicians, or engineers, or applied mathematicians, statisticians or scientists, and in society as it conducts its activities of commerce, construction, and regulation (Confrey, 1991).

In the United States, a reform movement in mathematics is underway. The many reasons for this reexamination and reform movement may be classified into five categories as indicated below.

1. Changes in mathematics and how it is used. New mathematics were discovered, and there was a resulting proliferation in the kinds and variety of problems in which mathematics is applied. The development of computers and the growth of computer applications increased significantly (National Research Council, 1991). These computer applications required the development of new mathematics in areas where applications of mathematics had not been feasible before the age of computers (Howson & Kahane, 1986).

2. Changes in roles of technology. Computers and calculators have not only changed mathematics, but how it is done (Rheinboldt, 1985). From kindergarten through the first 2 years of college, all mathematical techniques can be executed on a calculator. Consequently, the National Research Council (1991) addressed a rebalancing in the approach to all topics in school mathematics.

3. Changes in American society. American demographics have changed, causing different demands on the workplace for mathematics education (National Research Council, 1991). Johnston and Parker (1987) predicted that when today's children enter the work force, more mathematical skills will be required of these workers than in the history of mathematics education. Simultaneously, white males will represent a smaller fraction of new workers than in previous years (Oaxaca & Reynolds, 1988). Furthermore, a need exists for mathematics achievement to be ensured for all students (Office of Technology Assessment, 1988).

4. Changes in understanding of how students learn. The understanding of how students learn changed dramatically. Learning is not just a passive absorption of information, but students learn each new task with prior knowledge, assimilate the new information, and construct meanings (Resnick, 1987). This approach to learning is an active view of learning that is reflected in the way mathematics is taught (National Research Council, 1991).

5. Changes in international competitiveness. Many reports indicated that United States students did not attain as high a level of mathematics achievement when compared to students from other countries (Stevenson, Lee, & Stigler, 1986; McKnight et al., 1987; Stigler & Perry, 1988; Lapointe, Mead, & Phillips, 1989). The National Research Council (1991) warned that comparing the educational systems of different countries is dangerous because other industrial countries have different expectations and topics taught and the level of performance than in schools of the United States.

During the past 25 years, attempts were made to reshape and improve the mathematics curriculum, from international conferences and assessments to curricular experiments (Howson, Keitel, & Kilpatrick, 1981) to national investigations. These national investigations were conducted by the Committee of Inquiry into the Teaching of Mathematics in the Schools in 1982, the National Advisory Committee on Mathematical Education in 1975, and the Mathematical Sciences Education Board in 1989 (Romberg & Webb, 1993). These activities were directed by governmental agencies (federal, state, or local), commercial firms (for example, publishers and workshop entrepreneurs), or special interest groups (for example, mathematicians). Today in the United States, however, there are two major differences between past and present reform movement activities: (a) Leadership for movement is derived from professional organizations, and (b) consensus was reached among these organizations about the direction and the form the needed changes must take (Romberg & Webb, 1993).

After the publication of two major reports in 1983, A Nation at Risk and Educating Americans for the 21st Century, a need for change in our nation's schools in mathematics and science was supported. The basic argument was that mathematical and scientific understandings did not match its current or future needs. Subsequently, professional organizations called meetings. These reports generated reactions among professionals in the mathematical sciences, as presented in New Goals for Mathematical Sciences Education (Conference Board of the Mathematical Sciences, 1984) and School Mathematics: Options for the 1990s (Romberg, 1984). The first response report was a result of a meeting of the leaders in the mathematics community under the direction of the Conference Board of the Mathematical Sciences, which consisted of all the presidents of all of the professional mathematical sciences organizations; the second report was a result of a meeting of leaders in the mathematics education community directed by the National Council of Teachers of Mathematics (NCTM). Similar recommendations by both organizations were made, but each recommendation had a strategy by which to accomplish this. Recommendations 1 and 2 proposed guidelines for a K-12 mathematics curriculum (Romberg & Webb, 1993).

The NCTM, which at the time consisted of nearly 100,000 teachers of mathematics, mathematics educators, and mathematicians, carried out Recommendations 1 and 2. It organized the Commission of Standards for School Mathematics and published Curriculum and Evaluation Standards for School Mathematics, which responded to the concerns raised about the mathematical needs of our society for the 21st century (Romberg & Webb, 1993).

A draft of the 1987 version of the NCTM Standards was sent to each NCTM member during the 1987-1988 school year. Members reacted to the document with more than 1,000 comments. The thoughtful and helpful suggestions of mathematicians, mathematician educators, and classroom teachers strengthened the final document (Romberg & Webb, 1993).

The NCTM Standards (1989, 2000) were developed with new perspectives on school practices. This new emphasis was placed on creativity, innovation, problem-solving, and doing and thinking mathematically (Romberg & Webb, 1993). Hence, influences of constructivism guided the Principles and Standards for the Mathematics Curriculum (1989, 2000). For example, actively (rather than passively) is a key word because active is the basis for the NCTM Standards' principle of "actively building new knowledge from experience and prior knowledge" (p. 11). This principle was based upon the first principle of constructivism, which stated that the student actively is acquiring knowledge (Glasersfeld, 1989).

Constructivism's second principle stated that "the function of cognition is adaptive and serves the organization of the experiential world, not the discovery of ontological reality" (Glasersfeld, 1989, p. 182). Steffe and Wiegel (1992) supported the belief that the present mathematics reform will be successful if the first principle of constructivism is carried out with the second principle. Through action and interaction in the mathematics classroom, the teacher should be interacting by attempting to confirm, disconfirm, or modify hypotheses (Steffe & Wiegel, 1992).

The NCTM Standards were revised and a draft of the updated version was distributed in the fall, 1998. The revision process included input from the latest research supporting the NCTM goals. The first three Standards volumes were combined into one document. The basic goals were not changed.

These Standards addressed and challenged some teacher beliefs about mathematics and learning. Dossey (1992) found that the way one views mathematics has a major influence on one's teaching approach. Either a teacher is establishing environments to promote the development of mathematical thinking and connections in students, or a teacher is seeking for the best way to impart previously established knowledge to students.

Professional development that includes ongoing opportunities for teachers to share classroom experiences and work together to implement new methods is considered a necessary element of the NCTM Standards-based mathematics classroom (Huetinck & Munshin, 2000). Today's teachers may feel insecure dealing with mathematical topics in their students' textbooks that they have not experienced in their teacher preparation programs. One such example includes discrete mathematics' topic-networks. For this reason, professional development for teachers of mathematics must always include both content and pedagogy. More opportunities for such development exist today through regional district workshops, district workshops, university courses, professional conferences, and summer institutes for teachers (Huetinck & Munshin, 2000).

Good, Grouws, and Ebmeier (1991) found that teachers do make a difference in their students' classroom learning; Hammond and Ball (1997) found, to a stronger degree, that teacher expertise is the most important factor in determining student achievement: Teachers are the critical elements of successful learning. Furthermore, Hammond and Ball asserted that school reform cannot succeed unless it focuses on creating the conditions — including the curriculum contexts — in which teachers can teach well. Teacher expertise affects all the basic tasks of teaching. It affects what teachers understand, both about content and students, shapes how thoughtfully they select from texts and other materials, and how effectively they present material in class.

In the NCTM's Principles and Standards for School Mathematics (2000), in a discussion of "high-quality mathematics education" (p. 6), the Curriculum Principle stated, "the curriculum should offer experiences that allow students to see that mathematics has powerful uses in modeling and predicting real-world phenomena" (p.16). Furthermore, the Principles and Standards in the Problem Solving Standard for grades 6 through 8 explained that

Students can learn about, and deepen their understanding of, mathematical concepts by working through carefully selected problems that allow applications of mathematics to other contexts. Many interesting problems can be suggested by everyday experiences, such as reading literature or using cellular telephones, in-line skates, kites, and paper airplanes. (p. 256)

In addition, the Principles and Standards reported in the Problem Solving Standard for K through 12 that the curriculum should enable all students to build new mathematical knowledge through problem-solving and to solve problems that arise in mathematics and other contexts. These connections can be to other subject areas and disciplines in addition to students' everyday lives. Pre-kindergarten through grade 2 students should be building their mathematics experiences through the real world. Students in grades 3 through 5 should learn to apply mathematical ideas in other subject areas. But, in Grades 6 through 12, students should be using mathematics to explain complex problems in the real world (NCTM, 2000).

Principles and Standards (NCTM, 2000) declared that opportunity for students to experience mathematics in a context is important. Mathematics is used in social sciences, science, medicine, and commerce. This involved link is both content and process. For example, a year-long elementary school science activity involved designing instruments for measuring weather conditions and learning how to organize and communicate the resulting data (National Research Council, 1991). A multitude of connections to mathematics are relating and applying it to the real world.

Recent research on brain-compatible learning further supported connecting mathematics to real life. Neurobiological research showed that relevance is a biological function that results when an existing neural site makes a connection with a nearby site. When relevant content is presented, the connection is made; hence, if the content is irrelevant, the connection will not be made. The more links and associations the brain makes — determined in large part by the relevancy of the content — the "more neural territories are involved and the more firmly the information is woven in neurologically" (Jensen, 1998, p. 92). Through the process of patterning, groups of neurons "seem to be able to recognize and respond to meaningful learning" (Jensen, 1998, p. 95). Therefore, a curriculum that focuses on relevance and real-life context for learning is benefitting from what we know about brain-compatible learning to help students "make meaningful patterns and reinforce context" (NAHS, 1998).

Thus, authentic and appropriate content learning in which the mathematics curriculum is connected to real-world applications of knowledge reflected five standards, according to Newmann and Wehlage (1993). Authentic learning aims at a vision of authentic student achievement. Authentic distinguishes here between achievement that is significant and meaningful and that which is trivial and useless. These five basic premises for authentic learning include: (a) higher-order thinking, (b) depth of knowledge, (c) connectedness to the world beyond the classroom, (d) substantive conversation, and (e) social support for student achievement (Newmann & Wehlage).

In the premise of connectedness to the world, authentic learning is measured by the extent to which the class has value and meaning beyond the instructional context. For a class to gain authenticity, it must develop more connections to the larger social context within which students live. Instruction can demonstrate some degree of connectedness when children address real-world public problems; for example, in mathematics, clarifying a social or city issue by applying statistical analysis in a report to the city council on the homeless (Newmann & Wehlage, 1993).

In planning challenging projects, they are a means for students to acquire and apply knowledge to solve real problems, to produce real products, and to perform real tasks. Selecting and developing challenging projects requires careful attention to the unit's goals. Authentic projects must be planned with the expectation of advancing students' academic, technical, intellectual, and personal development (NASSP, 1998). For example, a mathematics teacher in a high school reported that she learned how to use projects to enliven and enrich her algebra classes by talking to every career and technical teacher in her high school. These interviews helped her create and develop some complex projects for her students. For example, her students ran a credit card business jointly with a business education class. As a result of students' involvement in such projects, students can understand how concepts are applied and connect between key academic concepts and career and technical applications. Through authentic learning experiences, real-world mathematical applications are supported. Hence, students value mathematics more and are able to use mathematics in everyday life (NASSP, 1998).

Another approach is the cognitive guided instruction in which the teacher builds on the mathematical knowledge of children according to what they already know. Cognitively guided instruction is based on the premise that the teaching-learning process is too complicated to prescribe. Consequently, teaching is problem-solving. Instruction consists of the teacher continually making decisions. But Carpenter and Fennema (1991) supported that helping teachers to make more informed decisions contributed to the most significant change in instruction. The main premise for this approach is that instructional decisions should be based on careful analyses of students' knowledge and goals of instruction. This premise requires that teachers have a thorough knowledge of the content, and they must be able to evaluate their students' knowledge of the content. Furthermore, this process requires teachers to understand distinctions between problems that are reflected in the solutions at different stages. Teachers must understand problem difficulty, as well as the different processes in solutions. They need to understand the different stages that students pass through in understanding concepts and procedures. The underlying inference is that instruction should be appropriate for each student. For each student, the understanding of the concepts and procedures requires that the learning should have meaning for each student. In this particular case, in order for learning to take place, the student will have to relate it to the knowledge that she or he already possesses. Carpenter and Fennema supported the principle that instruction should be organized to involve students in actively constructing their own knowledge with understanding. These authors also supported the principle that instruction should emphasize relationships among concepts, skills, and problem-solving.

In one example of a cognitively guided instructional program in the Madison, Wisconsin classroom of Annie Keith, the program specified that it did not provide a "scope and sequence" and did not provide instructional materials or activities for children. Rather, students spent time in a cognitively guided instruction class solving problems, using a variety of self-selected strategies and models, and talking about the solution strategies that they had constructed. The classroom environment was one in which peers and teachers respected each person's thinking. Students solved problems using a variety of different strategies representing different levels of conceptual growth. Consequently, different children solving the same problem were developing different levels of concepts and skills. The whole class could solve a number of problems, but the students often worked at centers where the teacher could adapt problems for small groups of children. The diversity provided a challenge, but it also provided an opportunity for children to experience a wide variety of strategies. Interactions among children using different strategies provided an opportunity for students to see relations among abstract and more concrete solutions, which provided a basis for them to develop understanding of abstract concepts and procedures (Hiebert et al., 1997).

Ms. Keith's instruction provided for a variety of opportunities for students to connect emerging concepts and strategies to concepts and strategies that they already understood. A main component of her class was the children continuing to contribute to the discussion of alternative strategies. Students used a wide variety of mathematical tools to solve problems, which in turn afforded a variety of solutions. The active participation of all students evidenced equity in the classroom. Respect was accorded to each student's thinking. The teacher's role was one of interaction with students, which required perception of each child's knowledge base. Ms. Keith participated with her students by listening to students and respecting what they had to say to the class (Hiebert et al., 1997).

In a historical retrospective, mathematics did not relate to everyday life, life in the real world. At the opposite end of the spectrum from cognitively guided instruction is traditional mathematics. Traditional mathematics as taught in the classroom is commonly associated with certainty, with knowing, with being able to get the right answer quickly (Heaton & Lampert, 1993). School experience shapes these cultural conjectures in which doing mathematics means remembering and applying the correct rule when the teacher asks a question. In traditional mathematics, mathematical truth is determined when mathematics and what it means to know it in school are acquired through years of watching, listening, and practicing.

Ruth Heaton, an experienced mathematics teacher, always considered herself a good teacher. Heaton had perceived mathematics as a "collection of rules and procedures to be learned and the answers to problems were either right or wrong" (Heaton & Lampert, 1993, p. 49). After Heaton observed Lampert, a teacher-educator of mathematics, in the classroom for 6 months, Heaton understood that mathematics was full of meanings, assumptions, and conditions. To Heaton, mathematics became more than the rules, procedures, and right answers (Heaton & Lampert).

Heaton remembered watching Lampert and students spend an entire class period on one problem, whereas in Heaton's traditional role as a mathematics teacher, students had solved 20 problems in a class period. Mathematics classes with Heaton's direct teaching had been the quietest part of the day, as the voice heard in the classroom was Heaton's own voice. Lampert's classes were laden with talk, discussion, and discourse, with Lampert as the facilitator (Heaton & Lampert, 1993).

Heaton was considered a traditional mathematics teacher prior to mathematics reform training. Lampert collected data about Heaton through discussions about the use of textbooks and teachers' guides and comments about how Heaton had followed manuals quite closely. In addition, Heaton had taught in a school district where performance objectives drove the mathematics curriculum, and goals primarily focused on the mastery of rules and procedures rather than conceptual understandings of mathematics (Heaton & Lampert, 1993).

Precision teaching, direct instruction, and mastery learning encourage good teachers to present topics in small, fragmented segments with easier skills preceding those more difficult to learn, whereas in this present mathematics reform, concepts are taught as an "integrated whole, not as isolated topics; the connections among them should be a prominent feature of the curriculum" (NCTM, 1989, p. 67). Cangelosi (1996) reported that some textbooks explained multiplication of fractions prior to addition of fractions because the algorithm was simpler than that for addition, in which one must bother with finding common denominators. Further, Cangelosi stated, this method of teaching confused students because of the unrelated algorithms. If addition of fractions was taught first, then multiplication of fractions presented, students were more likely to relate the two. Thus, concluded Cangelosi, the learning of addition of fractions enhanced the learning of multiplication of fractions.

Correctly applying the addition algorithm for regrouping from the ones column to the tens column is evidence of understanding in a traditional classroom. Similarly, in the traditional algebra class, producing the correct sequence of expressions through a succession of algebraic operations illustrates students' understanding. In the conventional geometry class, understanding a theorem implicitly means demonstrating a valid two-column proof.

There is no question that there has been mathematical teaching which required simply memorizing and reciting the words of some textbook, and there is likewise no question that such teaching is thoroughly bad . . . The aim of the work is to make the pupil master of the thought content and to enable him to apply it. Freed from the tyranny of memorizing, all the energies of the pupil are bent on thinking and what is once clearly thought through is already remembered. (Young, 1906, p. 331).

As early as 1906, at least one respected educator suggested the necessary break from traditional mathematics. J. W. A. Young (1906) in The Teaching of Mathematics strongly supported that "one needed many more tools and skills than just mathematical comprehension to be a successful mathematics teacher" (p. 331). The NCTM Standards' (1989, 2000) guidelines are rooted in Young's premises (Stein, 1993). He claimed that by using problems that interest and excite the students' curiosity, students are keen to develop important problem-solving strategies that can be applied in everyday situations, not just in the mathematics classroom or laboratory. Young felt "mathematics exemplifies most typically, clearly, and simply certain modes of thought which are of the utmost importance to everyone" (p. 331).

The NCTM Standards (2000), in the Curriculum Principle, purports that school mathematics curricula should focus on mathematics content and processes. Mathematics is considered important because of its utility in developing mathematical ideas, in linking different areas of mathematics, or in deepening students' appreciation of mathematics as a discipline and as a human creation" (p. 15).

The basic ideas in mathematics provide students with the ability to understand other mathematical ideas and link ideas across different topics of mathematics. It also should offer students experiences in predicting in the real world.

Real-world mathematics entails mathematical applications in everyday life. These applications prove to be noteworthy. Perceptions of real-world mathematics are explained in numerous sources. Roper (1994), from Great Britain, observed that "Mathematics is widely perceived as 'useful' in the 'real world, in everyday life', in one's present or future career and in the study of other subjects" (p. 174).

In support of this reasoning, a number of reports and studies emphasized the importance of real-world mathematics. Three fourths of the seventh- and eleventh-grade students in The Mathematics Report Card (Dossey, Mullis, Lindquist, & Chambers, 1988) reported that mathematics had a utilitarian use and was applicable in solving everyday problems. This practical use prevailed across gender and racial and ethnic groups. However, less than half of these same students anticipated working in jobs requiring mathematics. As the report indicated, it "is unfortunate that a majority of the high school students did not see mathematical skills and understanding as any part of their future work" (p. 99). This portion of the report implied a misunderstanding in students' minds about mathematics as it is taught and mathematics as it is applied in everyday work and life situations.

Real-world applications were emphasized throughout the NCTM Standards (1989, 2000) because the young student is more likely to remember mathematical applications related to the real world than those learned through a pencil-and-paper practice approach. The Standards assumptions for K through 12 affirmed the importance of the application of mathematics. For young students (K through 4) to be able to view mathematics as a "practical, useful subject, they must understand that it can be applied to a wide variety of real-world problems and phenomena" (NCTM, 1989, p. 18).

Understanding must occur for students to realize that mathematics is a perfect part of real-world settings and activities in other disciplines. The NCTM Standards (1989, 2000) for students in grades 5 through 8 emphasized the application of mathematics to real-world problems in addition to other settings relevant to middle school students. Some topics to receive increased attention in the high school mathematics curriculum included the use of real-world problems to motivate and apply theory in algebra; the use of real-world applications and modeling in geometry; and the use of realistic applications, modeling, and functions that are constructed as models of real-world problems in trigonometry. The common thread woven throughout the K through 12 curricula, according to the NCTM Standards (1989), is maximizing real-world problem situations to truly appreciate or understand the utility of mathematics in our future world.

Teachers attending a teacher's meeting on mathematics instruction at the Ford Foundation Urban Mathematics Collaboratives in 1987 commented on basic mathematical changes. One important conclusion was the need to stimulate students to reflect on real-world applications of mathematics and theory (Webb & Romberg, 1994).

To further confirm needs for real-world problem-solving, Debold's 1991 study found that the discrete operations construct of mathematics dominated students' thinking. DeCorte, Verschaffel, and Lasure (1995) confirmed that typical word problems in the standard textbooks did not activate students' knowledge about solving real-world problems. The traditional teaching of mathematics had resulted in students thinking of math as "a collection of symbol manipulation rules, and some tricks for solving stereotyped story problems" (Charles & Silver, 1988, p. 32). Students did not adequately link symbolic rules to math concepts in which students could link mathematical formalism to real-world situations. Students were programmed to find the facts and solve questions in answer to the typical word problem. Those same students, however, found it difficult to solve authentic problem situations in which they must address specific complexities in real-world mathematics.

Students need to be involved in the classroom environments in which word problems are not the routine word problems recognized in textbooks, but are of a real-world nature. Students need to find mathematics useful and discover that it can describe and manipulate real objects and real events. In presenting real-world problems to students, the primary factor in accommodating students' needs is to provide material that is interesting and meaningful and to show students how the material can be used in various situations (Midkiff, Towery, & Roark, 1991).

In trying to calculate teacher support for reform recommendations, Weiss (1995) sought mathematics teachers' opinions about the importance of effective mathematics instruction. Weiss used data from the 1993 National Survey of Science and Mathematics Education to assess teacher preparation, teacher pedagogical beliefs, and teacher perceptions of their preparation. On one report, mathematics teachers indicated that various strategies definitely should be a part of mathematics instruction. These mathematics teachers responded to "teachers believing that emphasis should be placed on solving real problems" (p. 9) with 80% of grades K through 4 teachers agreeing. In grades 5 through 8, 78% of teachers agreed, and in grades 9 through 12, 57% of teachers assented.

Table 1 reports six real-world mathematical applications, representing engaging classroom activities that incorporate age-appropriate mathematical concepts in everyday contexts. DeBruin and Gibney (1979) created a real-world mathematical application that included determining the best brand of paper toweling for a given situation. Students were first introduced to the basic process skills and then began their study by selecting five brand names of paper toweling: Brand A, Brand B, Brand C, Brand D, and Brand E. They used process skills to engage in hands-on activities to determine the best brand of paper toweling for a situation. Students used process skills of observing, estimating, approximating, using numbers, measuring, space-time relationships, inferring, and manipulating equipment. In addition, students read, interpreted, and constructed charts, tables, and graphs of the data collected.

Table 1 Summaries of Real-World Mathematics Curricula
Author (Year)	Paper	Summary
DeBruin & Gibney (1979)	Solving everyday problems using science and mathematics skills	Real-world activities to determine the best brand of paper toweling for a given situation
Battista (1993)	Mathematics in baseball.	Describes topics in baseball and ways to use them as problems
Vatter (1994)	Civic mathematics.	The fundamental goal of this curriculum is to teach mathematics skills in the context of important social issues of widespread concern to teenagers
Lobato (1993)	Making connections with estimation.	Problems on estimation are put into real-world contexts.
Wagner (1979)	Determining fuel consumption.	Applying mathematics to real-world problems.
Hynes (1998)	Mission Mathematics.	Mathematical problems focused on NCTM curriculum standards in the context of aerospace activities.

To connect students in mathematics to the real world, Battista (1993) developed a project and study of mathematics in baseball. Statistics is required in the rules of baseball in which statistics is kept on all players and teams. A part of player statistics is calculating the batting average. Discussions can be held with leading questions, for example: "Who has the best batting average? What does the batting average tell you about a batter's performance? About how often would you expect a .300 hitter to get a hit? Why? Another batting statistic is the slugging percentage. Statistics also included the won/lost record for pitchers and team standings.

This baseball project included geometric problems that were investigated. These geometric problems encompassed geometric constructions, Pythagorean theorem, field sizes, pitcher's speed, spheres and rectangles, and volume, slope, and the pitcher's mound. Computers could be used to extend the study of several of the concepts. For example, students could be asked to construct computer spreadsheets (Battista, 1993).

This project not only promoted the content strands of the NCTM's curriculum standards, but it provided for the development of the process strands of problem solving, communication, reasoning, and connections. This project forced students to use mathematical ideas in context (Battista, 1993).

In Ithaca, New York, a civics mathematics curriculum was developed as a mathematics-social studies interdisciplinary course in which social issues provided the context for problems. For example, a student developed a project on homicide, which is of tantamount concern for teenagers. The course employed current data and events and it could be constantly revised. Data could not be more than a few years old, and subsequent revisions could be introduced when time permitted. The project required a student to: (a) ask relevant questions, (b) research it, (c) answer it in prose with data analysis to support the answer, (d) present the results in an oral presentation to the class, and (e) display some appropriate part of it in school or in the community. Additional examples of topics included water resources, longevity, family structure, employment and income, immigration, poverty, homelessness, housing, banking, wealth, demographics, global warming, education, career, nutrition, substance use and abuse, voting, and military service (Vatter, 1994).

Making connections with the real world helps students see the usefulness of mathematics and understand how particular topics are applied (Lobato, 1993). An example of making these connections is between students' understanding of numbers and extensions of those concepts to estimating. Estimation activities can reinforce connections with mathematical topics, other disciplines, and the real world. For example, figuring out the number of students it would take to make the height of the Pyramid of Cheops. Another activity might use pictures of flags; students would estimate the percent of a flag that is devoted to a specific color (Lobato, 1993).

Wagner (1979) stated that our students need to understand the process of applying mathematics to "real life" problems in order to cope with our technological society and to perform their own personal mathematical applications in everyday life. Wagner developed a unit on automobile fuel consumption 22 years ago by asking these primary questions that are just as valid today as they were then: (a) Should I buy a Sprint automobile having average gasoline mileage, or should I pay an extra $500 and buy a Spartan automobile having superior gasoline mileage? (b) How much gasoline will our country save by improving the fuel efficiency of our cars?

Students must process information by asking five questions: (a) What specific additional information must we know in order to choose between the Sprint and the Spartan? (b) How can we translate the problem into mathematical terms? (c) Before solving the problem, what assumptions have we made so far? (d) What is the solution? (e) Can we generalize these results? The same process can be applied to both primary questions on fuel consumption. Applications of mathematics depend on the considered assumptions and rational judgments of human beings (Wagner, 1979).

Mission Mathematics (Hynes, 1998), a book of aerospace activities, was developed primarily to focus on the NCTM curriculum and evaluation standards in the context of aerospace activities. Another goal was to engage students actively in NCTM's four process standards: (a) problem solving, (b) mathematical reasoning, (c) communicating mathematics, and (d) making mathematical connections among topics in mathematics to other disciplines and to real life. A third goal was to translate the work of engineers and scientists at NASA into language and experiences appropriate for young learners. Goal four was to provide teachers with mathematics activities that can complement many of the available NASA resources for students. The book was divided into four chapters: (a) aeronautics, (b) human exploration and development of space, (c) space science, and (d) mission to planet Earth (Hynes, 1998).

In the chapter on aeronautics, mathematics concepts included area, picture graph, pictograph, bar graph, measuring distance, range, median, mode, stem-and-leaf plot, mean, line plot, multiple-bar graph, fractions and decimals, circle graph, plotting points on a coordinate grid, probability, and time. The levels were identified and provided for in each chapter, along with "Class Conversation" followed by a set of questions. Through these discussions, students were helped in relating previous experiences to the activity, focusing on the mathematical task or problem, identifying and exploring patterns and relationships, applying problem-solving strategies, and discovering connections to other mathematical concepts or other cross-disciplinary tasks (Hynes, 1998). The common thread woven throughout these six real-world mathematical applications curricula was the practical and useful side of the problems.

Applied science and mathematics educators allege that mathematical literacy requires skills of problem-solving and critical thinking, which are in demand in the workplace. Students also need to be taught to plan and be able to make sense of statistics and other data. Problems requiring these skills should be presented to students (Education Writers Association, 1988).

A conflict between the theory of mathematics and its "close connection to the real-world gives rise to many problems in mathematics education" (Willoughby, 1990, p. 12). Willoughby observed that some teachers believed that mathematics should be taught entirely in an abstract way, whereas other teachers ascertained mathematics as closely related to the real world in which real-world problem-solving could be extremely useful. These latter teachers agreed that all mathematics should be "learned, taught, and discussed in connection with some physical referent — never relying on totally abstract symbols" (p. 12). The connection between the real world and mathematics should be shown for the student to fully appreciate "the power and beauty of the mathematics . . . in its abstractness" (p. 13).

Mathematics should be taught in connection with other disciplines to promote understanding of the changing natural phenomena around us, advocated Young in 1906 at the University of Chicago. Young thought that geometry, the algebras, and other courses should be taught simultaneously throughout the four years of high school, rather than in tandem as distinct subjects. Furthermore, the NCTM Standards (1989, 2000) emphasized the need for a wide spectrum of topics to be included in this curriculum. For example, number concepts, computation, estimation, functions, algebra, statistics, probability, geometry, and measurement should be taught through connections to real-world problems and within the mathematics domain.

Echoing Young's (1906) thoughts from early in the 20th century, NCTM (1989) pronounced, more than 80 years later, that mathematics "should be taught as an integrated whole, not as isolated topics; the connections among them should be a prominent feature of the curriculum" (p. 67). As students study one topic, "relationships to other topics can be highlighted and applied" (NCTM, p. 67). For example, a lesson in measurement can be an opportunity for students to plan and solve problems while discovering and studying metric, geometric, and algebraic ideas. The NCTM Standards (1989) asserted using "a mathematical idea to further their understanding of other mathematical ideas" (p. 84). The Professional Standards for Teaching Mathematics (NCTM, 1991) advocated teachers engaging students "in a series of tasks that involve interrelationships between mathematical concepts and procedures" (p. 89). Furthermore, content learned in an isolated manner does little for students' abilities to learn the content as it is not presented in a meaningful way.

Shealy (1993) also believed that teachers needed to connect mathematics to real-world and applied situations, maintaining that, because students need to move from the real world to a conceptual model, they must have a strong understanding of the real-world problem domain and the mathematical domain related to it. DeCorte et al. (1995) validated this by demonstrating in their study that students in general tended to exclude real-world knowledge and realistic considerations when confronted with the problematic versions of the problems. They found that students were resistant to solving mathematical problems in a practical way in considering all possibilities. In other words, students did not extend and make connections in their minds because they were too programmed by the routine-type word problem. This study supported the need for students to solve real-world problems in mathematical modeling. To substantiate this, only 2 of 48 fifth-grade students in Debold's 1991 study recognized the power of mathematics to help them solve problems in the real world. But, in this same study, two fifths (40%) of the students responded by defining mathematics as useful. One child iterated, "It's an interesting thing for you to do and, you know, um when you really grow up . . . you're gonna have to . . . use math." Evidence from this study supported problem-solving dominated by young students' experiences with classroom arithmetic. In addition, Debold concluded that emphasis on the daily mathematics curriculum on real-life problem-solving resulted in changes in students' constructs of mathematics.

These interrelationships or connections within mathematics should be used within the study of mathematics consistently. The geometry teacher's goal is to teach about mathematical systems and demonstrate how we can derive various theorems from a small set of axioms. A much smaller system than all of the Euclidian geometry would be far more appropriate, so that students can truly see the connections. But, even when working within the geometry system, the mathematician must go outside this system to retrieve learned information, such as arithmetic or algebra (Willoughby, 1990). Patterns between the numbers of faces, edges, and vertices of the solids ought to result in conjecturing a theorem about the relationship that would be written using algebra; this relationship is classified as topology.

Yet in another way, in an operational sense of addition, Slavit (1995) found connections between specific kinds of knowledge of arithmetic and the student's ability to model the actions of a problem. Slavit maintained that operational sense promoted deep understandings of the operation involved in various kinds of flexible conceptions that the learner can interrelate. Specific aspects of operational sense involved counting, properties, relationship to other operations, knowledge of symbol system, and use on abstract goals. Implications were made about additive knowledge that affects problem-solving behavior. Slavit concluded that some connection existed between types of knowledge associated with addition and the types of strategies students used. The knowledge of specific properties allowed the students to better model the actions present in the tasks.

During the child's early formative years in learning, understanding, and realizing, these connections will help the student relate to other mathematical connections in his or her later school years. Connections referred to as vertical connections, or spiral connections, existed throughout the grade levels. These connections are harder to evaluate and make happen. An example is the study of functions (Willoughby, 1990).

Rosenfield, an expert teacher, wanted "students to become fluent users of the powerful tools of mathematics" (Peterson, Putnam, Vredegoogd, & Reineke, 1991, p. 16). Rosenfield emphasized that main concepts should be interwoven within rich mathematical activities, rather than taught as discrete concepts or subskills. This teacher further argued that learning virtually always proceeded best by going through the concrete, connecting, and symbolic levels. In this same study by Peterson et al., another expert teacher, Rodriguez, viewed connections as necessary to conceptual understanding in mathematics and attempted to help students make connections to the various contexts of use of mathematical concepts and procedures.

For example, the Texas Education Agency (TEA), in its Texas Developmental Math Program, approached improving mathematics performance by the inclusion of connections within mathematics.

Geometry can be used to solve problems involving patterns or probability; statistics skills can be used to understand measurement problems; number sense and place value help to develop understanding of computational algorithms. Mathematics should be valued not as a set of isolated unrelated topics, but rather as an interesting web of interrelated concerns and understandings. (TEA, 1989, p. 8)

In 1906, J. W. A. Young wrote that mathematics courses should be taught in conjunction with other similar and related mathematical courses, but not taught as an isolated subject within the category of mathematics. Thinking in the field of mathematics education is finally coming into accord with Young's ideas that, through real-world mathematics problem-solving, connections and integration of mathematics within the domain and with other disciplines are introduced. Integrated Mathematics

New mathematics textbooks were adopted for grades K through 5 during school year 1998-1999 in Texas. Changes in textbooks included integrating the various disciplines with mathematics and integrating mathematical applications within mathematics. The United States is one of the last countries in the world to teach the subjects separately (Driscoll, 1988).

The integration of writing in mathematics assisted teachers in finding reasons why students failed to make connections between strands of the mathematics curriculum. Students sometimes do not connect their previous concepts with what they are currently doing. For example, they may not recognize that operations applied to whole numbers work the same way as the same operations used with fractions. If teachers look only at the computation of the problem, they may miss students not making connections. When students write about those relationships, the connections become more obvious (Resnick & Ford, 1981).

In a course on mathematics methods for elementary teachers, students were asked to write a "story problem" that included the division of two fractions (1/2 divided by 1/4 = 2). Interestingly, only a few students were successful until it was suggested that they think about the meaning of division when used with whole numbers. Then, the students were able to construct a satisfactory question. This response from one student was in many ways typical of others in the class. She wrote:

If you divided one half of a piece of pie into 4 equal areas, how many half pieces would it take? If you have 2 sets of 1/4 piece of pie, how much of a pie would you have? How many groups of 1/4 do you need to make 1/2? (Drake & Amspaugh, 1994, p. 45)

In less than a year, this college-educated adult was to be a classroom teacher. If prospective teachers fail to make the connections between the meaning of division of whole numbers and the division of fractions, it is no surprise that their students will have difficulty making those same connections. Unless teachers know that relationships among the strands of the curriculum are not clear, no instruction will occur to help delete misconceptions (Drake & Amspaugh, 1994).

In the 1993 National Survey of Science and Mathematics Education, mathematics teachers throughout the United States were surveyed to collect data on science and mathematics course offerings, enrollment, background preparation, textbook usage, instructional techniques, and use of science and mathematics facilities. The survey results revealed that 65% of grades 5 through 8 mathematics teachers agreed on the integration of mathematics subjects, whereas 73% of grades 5 through 8 teachers agreed on the integration of mathematics with vocational-technology education (Weiss, 1995, p. 10).

Roper (1994) stated that an integrated curriculum required a good understanding of many fields of knowledge, their different sociologies and natures, and a willingness to be prepared to work against a background of uncertainty that may need continuous resolution. He advocated thinking of mathematics as a means of communication, something to represent, explain, and predict. Conclusions were drawn that included the need for mathematical modeling. For example, the Mechanics in Action Project (MAP) was instrumental in promoting mathematical modeling and real problem-solving. This successful project's aim was for the students' perceptions of mechanics to change. Some innovative alterations included curriculum audits in which schools probed where and when in the curriculum particular topics are taught (Roper, 1994).

In Great Britain, other changes for the integration of mathematics have occurred. Schools began holding project or theme weeks in which entire groups of students were involved in looking at a common theme in all subject areas for one year. Roper (1994) proposed that an interpretive, reflective, and referential view of mathematics appeared to be one which would be vital to mathematics and to its use in other subjects.

Great Britain's national educational system provided for a National Curriculum (NC) in which two main points were observed. The NC affirmed and initiated the integration of mathematics into the wider curriculum and the relationship between mathematics and other subjects. Very little, if any, practical help was offered and few exemplars were given (Roper, 1994).

However, the United States's NCTM addressed the integration of mathematics with other disciplines through new curriculum programs and new textbooks, such as Mission Mathematics (Hynes, 1998). This program integrated science and other disciplines and applied problems in real-world situations. One of Mission Mathematics' goals was to "show contemporary applications of mathematics in an important context and using contemporary methodologies" (Hynes, p. 7). The program encouraged students to appreciate mathematics in all occupations and fields of study.

In the American Association for the Advancement of Science's Projects 2061 report (Keynes, 1997), a model for problem-solving using statistical principles — prediction, model, uncertainty (careful examination), and optimization (finding maximum and minimum) — was developed. This model was used in different types of curricula. In the pursuit of thematic-based curricula, it would certainly require teachers educated in and practicing a discipline-based approach to be educated in what is valuable in mathematics. The sciences may be our best allies in this approach. The curriculum suggested in Projects 2061 removed rigid boundaries between traditional disciplines and developed underlying themes and the "big picture" (Keynes, 1997).

Real-world mathematics requires that mathematics be learned through making connections within and outside of the domain and by integrating mathematics with other disciplines. Through learning the structure of mathematics, students realize and understand the power of mathematics in their everyday world.

Mathematics in the 20th century evolved from traditional mathematics to linking mathematics more to real-world mathematics. The curriculum and teachers' methodology developed into the current mathematics reformation.

The encompassing of the curriculum and the teacher is the classroom environment. Classroom environment is necessarily a multi-faceted topic. Table 2 summarizes studies reviewed, all of which showed an increase in student achievement resulting from the implementation of real-world applications in mathematics programs. No studies were located with findings that real-world mathematical applications resulted in lower achievement or no increase in achievement in comparison to a traditional curriculum. These studies tended to emphasize one of three different, but interconnected, facets of the classroom environment: (a) the teacher as educational resource, (b) the curriculum, and (c) teaching materials.

The Teacher as Educational Resource

While each of the studies discussed in this section contains components relating to teaching materials and curriculum, their conclusions were specifically addressed to the professional development of the classroom teacher. Each of these studies addressed the teacher as an important educational resource in the same sense as well-thought-out lesson plans and carefully crafted instructional materials.

Table 2 Studies Showing an Increase in Student Achievement Resulting from the Implementation of Real-World Applications in Mathematics Programs
Study	Type	Instrumentation	Participants
The Teacher as Educational Resource
Good, Grouws, & Ebmeier (1991)	quantitative	SRI Test (pre & post) problem-solving posttest	elementary and secondary students
DeCorte, Verschaffel, & Lasure (1995)	quantitative	10 pairs of problems	75 fifth-grade students
Connell, Peck, & Buxton (1992)	quantitative	Valley Crest Mathematics	NA
Peterson, Putnam, Vredegood, & Reineke (1991)	quantitative	Survey	493 students
Steele (1995)	qualitative	Descriptive, interviews, observations	1 fourth-grade class
Anderson (1987)	quantitative	Initial Student Question-naire, Initial Teacher Questionnaire, Final Student Questionnaire, Final Teacher Question- naire, Student Aptitude Test, Classroom Rating Instrument: Opportunity to Learn
The Curriculum
Cobb, Wood, Yackel, & Perlwitz (1992)	quantitative	Achievement test ISTEP Project Arithmetic Test	200 students
Drager-McCoy (1996)	quantitative	Survey	459 teachers
Fuson, Carroll, & Drueck (2000)	quantitative	Whole class tests by researcher from Northwestern University longitudinal study	392 second graders
Huntley, Rasmussen, Villarubi, Sangtong, & Fey (2000)	quantitative	100 questions covering advanced algebraic ideas, relationships, and techniques	540 - 1,000 advanced algebra students
Teaching Materials
Pascale (1979)	quantitative	Subtest M-2 of Iowa Test of Basic Skills	117 sixth-, seventh-, and eighth-grade students
Manouchehri (1998)	quantitative	Questionnaire	51 middle school teachers from 10 school districts

Active teaching helps students comprehend what they are learning, stated Good, Grouws, and Ebmeier (1991). Based upon earlier naturalistic findings and experimental research, these three researchers developed a systematic program of instruction that included:

1) instructional activity is initiated and reviewed in the context of meaning; 2) students are prepared for each lesson stage to enhance involvement and to minimize errors; 3) the principles of distributed and successful practice are built into the program; 4) active teaching is demanded, especially in the developmental portion of the lesson (when the teacher explains a concept being studied, its importance, etc.). (p. 12)

As a result of research they conducted on classroom mathematical processes, Good, Grouws, and Ebmeier (1991) maintained that teachers make a difference in the classroom learning of individual students and asserted that the concept of active teaching is useful for studying classroom teaching and learning. After their first experiment in 1979, they decided to proceed with Experimental Study II in elementary schools. Verbal problem-solving was studied to determine if teacher behavior and student achievement could be affected. These researchers felt that, if mathematical knowledge is to be applied to the real-world, students needed practical problem-solving skills. Because of the scant material available, instructional strategies were developed by writing a training manual with instructional strategies for teachers to teach verbal problem-solving. Three organizational patterns were represented in this teacher sample: (a) semidepartmentalized structure, (b) some teachers taught only mathematics, and (c) some teachers were in open classes in which team teaching and individualized instruction were used. The raw means and standard deviations for the SRA (pretest and posttest) and the problem-solving posttest, by treatment condition and by organizational structure, reflected student performance improving from pretests to posttests in all cases on the SRA test. In addition, all treatment groups outperformed the equivalent control groups. Two of the three treatment groups had higher mean performances than equivalent control groups on the problem-solving test. An analysis with adjusted mean scores was completed on the problem-solving test that used the pre-SRA test as a covariant to compare the significance of adjusted means across all treatment and control classes. This analysis demonstrated that the performance of the treatment group surpassed that of the control group in a way that approached significance. The experimental program had positive effects upon students' problem-solving skills (Good, Grouws, & Ebmeier, 1991).

This research was expanded to include secondary schools. The most important finding for this research was that instructional time spent on problem-solving activities did correlate significantly with students' problem-solving achievement scores. Students' performance in problem-solving in both partnership and treatment classes was superior to problem-solving performance in control classes. As a result of their research, Good, Grouws, and Ebmeier (1991) concluded that teachers represent a distinct educational resource and that money invested in attracting, educating, and retaining capable teachers can result in increased student achievement.

DeCorte et al. (1995) studied the lack of activation of real-world knowledge in elementary school students' understanding and solution of school arithmetic word problems. These researchers concluded that a need existed for more authentic problem situations that would stimulate students in their development of real-world mathematical skills and that curriculum designers needed to pay explicit attention to the complexities involved in realistic mathematical modeling. The authors' second recommendation was that the way in which mathematics is taught should reflect a constructive, active, and collaborative way of learning; and, third, special attention should be given to the establishment of a new mathematics classroom culture by explicitly negotiating new social norms about what counts as a good mathematics problem, a good solution procedure, a good response, and by renegotiating the role of the teacher and the students in a mathematics class.

Furthermore, Connell, Peck, and Buxton (1992) attested to the belief that teachers need to become aware of and members of the real-world culture of mathematics. After the training of teachers with instructional support, these teachers became indoctrinated into the culture of real-world mathematics. The following characteristics were observed about the teachers' change in characteristics: (a) Instruction became student centered and constructivist in nature; (b) the instructor's role became that of question asker and problem poser; and (c) problem-solving, persistence, and resourcefulness on the part of the students became highly valued. Consequently, students' growth in mathematics was statistically significant. It was in the areas of Extended Mathematics (pre-Algebra) Problems, Miscellaneous Problems (which required a variety of problem-solving strategies), and Estimation (which, although not formally discussed, was inherent in all student work) that the greatest increase in student performance was observed. The near doubling in student performance in each of these areas provided strong evidence that the instructional emphasis upon problem-solving was effective for this group.

A survey as part of the work of the Center for the Learning and Teaching of Elementary Subjects at Michigan State University was conducted in 1988-1989. Its purpose was to obtain a picture of elementary teachers' current practice in six subject areas: (a) mathematics, (b) science, (c) social studies, (d) literature, (e) art, and (f) music. Specifically, it identified teachers' reported use of specific curricula and materials designed to promote greater student understanding, problem-solving, or thinking in each subject area and teachers' judgments of how knowledgeable and effective they are in each subject compared to other elementary teachers (Peterson, Putnam, Vredevoogd, & Reineke, 1991).

Elementary teachers in three states — California, Michigan, and Florida — were surveyed. These three states were chosen to participate in the survey because they represented different policy contexts and, at the state level, they differed significantly, both substantively and procedurally, in their approaches to subject area curriculum guidelines and policies. In addition, the survey was administered to three elementary teachers who had been identified as experts as part of another study. Each teacher was asked to prepare a written document addressing three mathematical goals, listing important understandings related to each goal, and developing a scenario for teaching one of the understandings at each of two grade levels, second and fifth. Each expert teacher went to the Michigan State University campus, where a 5-hour interview and a 6-hour interview were conducted during the summer of 1988. From these interviews and documents, researchers were able to construct a profile of each expert teacher (Peterson et al., 1991).

For example, teacher Dolly Davis believed that mathematics should be fun, easy, useful, and relevant to students' lives now and in the future. Davis designed her classroom activities to capitalize on familiar contexts, and she created activities through which students could see the applicability and interrelatedness of all they learned. Her students created a small town that they built, designed, and participated in throughout the school year. Mathematical problems were introduced through their town. Students punched a time clock when they began and left work at their jobs in the town. Students were paid 3¢ to 5¢ per minute. Subsequently, each student had to calculate the amount of money that he or she earned on the job during the week. During the designing of the town, Davis had the children create scale drawings of the actual buildings in the town. In these mathematical activities, students encountered mathematical problems and ideas in which they applied what they had learned in mathematics lessons (Peterson et al., 1991).

Of importance to another teacher, Elaine Rosenfield, was that mathematics should make sense to students; mathematics must connect in meaningful ways with what children already knew. She wanted students to become natural users of the powerful tools of mathematics. Her beliefs included viewing mathematics as "a tool to organize information and to make decisions about real problems" (Peterson et al., 1991, p. 16).

The third expert teacher, Yolanda Rodriguez, generated three major goals for her students. First, she wanted students to become empowered by thinking mathematically. Second, she wanted students to think of mathematics as a useful tool in that they knew when and how to apply mathematical skills in a variety of contexts and would be able to make connections between their mathematics knowledge and real-world and scientific contexts. Her third major goal was to exemplify a sense of wonder and sense-making in mathematics that would compel her students in the same way (Peterson et al., 1991).

Steele (1995) affirmed that mathematics textbooks should not contain page after page of procedural computations but should contain significant problems central to primary concepts in mathematics. Also significant was Steele's finding that the teacher must listen closely to students when they describe their thinking or tell how they solve problems. This behavior on the part of the teacher provides a careful-listening model for students to follow.

Students' perceptions of their classroom and the instruction they receive influence their achievement and attitudes, generalized Anderson (1987). Student perceptions of the task orientation of their classrooms indicated a consistent influence on student achievement and attitude across countries. Additionally, student perceptions of the degree to which their teachers provided the necessary structure for their learning provided consistent influences on student achievement.

These studies indicated that the teacher played an important role in the communication of the curriculum to the students. Recent research furthermore showed that teachers' content and pedagogical content knowledge influence how they teach and evaluate the content (Ball, 1988, 1990; Borko et al., 1992; Carlson, 1988; Henderson, 1995; Shulman, 1986; Shulman & Grossman, 1988).

Curriculum

Cobb, Wood, Yackel, and Perlwitz (1992) found that a problem-centered instructional approach to mathematics was superior to a traditional curriculum. In this study, 5 second-grade classes in 2 schools participated in a project that was generally compatible with a constructivist theory of knowing. At the end of the school year, students in these classes and their peers in 6 non-project classes in the same schools were assigned to 10 textbook-based third-grade classes on the basis of reading scores. The two groups of students were compared at the end of the third-grade year on a standardized achievement test and on instruments designed to assess their conceptual development in arithmetic, their personal goals in mathematics, and their beliefs about reasons for success in mathematics. The levels of computation performance on familiar textbook tasks were comparable, but former project students had attained more advanced levels of conceptual understanding. In addition, they held stronger beliefs about the importance of working hard and being interested in mathematics along with understanding and collaborating. Further, they attributed less importance to conforming to the solution methods of others.

In the implementation of an applied mathematics curriculum, Drager-McCoy (1996) found that 89.1% of 91 teachers helped students understand real-life uses of mathematics. Comments from 60 teachers, who recommended using the applied mathematics curriculum, collectively advocated, "Curriculum provides a hands-on approach with real world connections" (p. 82).

Fuson, Carroll, and Drueck (2000), reporting on two studies of the effect of the Everyday Mathematics (EM) curriculum, showed positive results for this approach. To assist students, mathematical ideas were often presented in real-life contexts and in problem-solving activities. A greater opportunity to learn, in terms of both total time and the inclusion of more ambitious topics, was accompanied by activities that made the mathematics meaningful to the students. On a test of number sense, the heterogeneous EM grade 2 students scored higher than middle-class to upper-middle-class United States traditional-textbook students on two items and matched them on the remaining items. Their scores were equivalent to middle-class Japanese students. On a computation test, the grade 2 EM students outperformed the same United States students on three items involving three-digit numbers. However, Japanese children outperformed them on six of the most difficult test items. EM third-grade students scored higher on items apprising knowledge of place value and numeration, reasoning, geometry, data, and solving number stories. On a few questions in these areas, EM grade 3 students even scored as high as or stronger than seventh graders taught with a traditional approach. Fuson et al. concluded that teachers can learn to support such learning through use of an appropriately structured and developed curriculum.

A carefully developed curriculum called the Core-Plus Mathematics Project (CPMP) was compared to a more conventional or traditional curriculum on growth of student understanding, skill, and problem-solving ability in algebra. The National Science Foundation funded the CPMP in 1992 to construct a 3-year integrated mathematics curriculum for all students, plus a 4th-year course continuing the preparation of students for college mathematics. Its main goal was to construct, implement, and evaluate a high school mathematics program illustrating principles and practices recommended by the recent reform. The curriculum materials focused on real-life and mathematical contexts and investigations that led students to important mathematical understandings and skills (Huntley, Rasmussen, Villarubi, Sangtong, & Fey, 2000).

Results from this comparison study indicated that the CPMP curriculum was more effective than the traditional curriculum in developing student ability to solve algebraic problems when those problems were presented in real-world contexts and when students used graphing calculators. The most consistent finding of algebra assessments in this study was that students learned more about topics that were emphasized in their mathematics classes and less about topics that were not emphasized. The content of curriculum text materials and classroom coverage of those materials made a difference (Huntley et al., 2000).

Teaching Materials

Pascale (1979) recognized the need for the mathematics curriculum to include real-world investigations. Because of this belief, Pascale developed, piloted, field tested, and revised multi-faceted learning kits designed to help students (remedial through gifted) improve their skills and attitudes and awareness and appreciation of the use of mathematics in the real world. The kit provided games, small- and large-group-oriented strategies, simulations, and practical applications of real-world mathematical skills. The summative focus of the evaluation was the pretest-posttest assessment of student performance in mathematical concepts and problem-solving and attitude and interest toward mathematics as measured by the Iowa Test of Basic Skills and Attitude/Interest Towards Mathematics Scale. Year 1 findings indicated that exposure to kits developed from student need and interest produced statistically significant gains by the experimental groups on both mathematical problem-solving and attitude toward mathematics. The results specifically indicated the efficacy of the project-developed materials in producing significant gains in students' mathematical problem-solving skills as measured by the ITBS M-2 subtest.

Little research on teachers' interactions with creative and NCTM Standards-based curricula existed previously. Consequently, 51 middle school mathematics teachers from 10 school districts in the St. Louis, Missouri area conducted a study during the 1995-1996 and 1996-1997 school years to gain insight into the process and evaluation and development of four NCTM Standards-based curriculum programs (Manouchehri, 1998). These teachers participated in the Missouri Middle School Mathematics Project-M3, which was a curriculum-review project funded by the National Science Foundation. This project provided for the participating teachers to review and evaluate their Standards-based middle level (grades 6 through 8) curricular materials in schools. During this evaluative process, a profile of factors emerged to enhance or impede middle school teachers' use of the programs. All teachers began the implementation of the programs with enthusiasm, but all teachers did not sustain this enthusiasm for the duration of the programs. Less than half the teachers (21) reported increased student interest in learning mathematics and greater involvement in class activities as a result of the Standards-based materials. Teachers familiar with and accustomed to constructivist approaches demonstrated more enthusiasm for the programs. These teachers found the programs intellectually stimulating and practical. But the more experienced teachers, who had continued to use the traditional approach, found the programs insubstantive and viewed this program as an affront to the strategies, methods, and materials they had developed and used for some time.

Teachers found that materials did not provide teachers with guides to think about the long-term dimensions of curriculum construction and implementation. Most of the teachers were left alone to reflect and decide how to connect ideas between and among the various activities and units. Many teachers evaluated the materials as not providing them with substantial understanding about how to deal with student thinking (Manouchehri, 1998).

The first mention of the word gifted appeared in 1644. The word "gifted" meant "endowed with gifts," or "talented," which implied a person having a special talent for public speaking: "It is one thing to say a gifted man may preach, but another thing to say a ruling elder may do it" (Oxford English Dictionary, 1981). In 1677, the term was defined as able to do something well; for example, women were referred to as gifted in sewing and knitting. In the 19th century, a reference was made to the term connected with genius.

Throughout the 20th century, giftedness evolved into a theory-based definition. Cohen (1988) outlined the current theories of giftedness by advocating that a theory should consider: "(1) the nature of the gifted child and (2) the education and identification of the gifted child" (p. 29). Gardner's multiple talents theory, Sternberg's triarchic theory, and Meeker's structure of the intellect gained in importance by adding knowledge to the term "giftedness." In addition, Dabrowski's theory of emotional development, Clark's brain-mind theory, Renzulli's theory, and Tannenbaum's theory contributed to our knowledge of giftedness — what it is, what it means to be gifted (Piirto, 1994).

Piirto (1994) commented on Feldman's 1991 paradigm for giftedness that was emerging to emphasize qualities rather than traits. This new paradigm was primarily based on individual excellence; giftedness was defined by the context in which the gifted individual finds himself or herself. The old definition was test driven; the new definition was achievement oriented: Giftedness is a developmental process. This new, emerging definition of giftedness was collaborative in that it was determined by consultation, not by schools. A person was gifted in a domain or in something that qualified the domain as diverse (Piirto, 1994).

The Gifted African-American Student

Historically, gifted students have been overlooked. This is particularly true of gifted minority students. Gifted African-American students have special needs and special characteristics as students in all cultural and academic groups do. Davis and Rimm (1998) reported giftedness for the culturally different student is obscured because of cultural bias in test instruments and other identification methods. However, 38 states in 1998 had specific instructions from their state departments of education to include the special needs of the identification of gifted minorities and culturally different students (Davis & Rimm, 1998).

Davis and Rimm (1998) reported that a major problem in identifying poor and minority gifted students was from the attention on homogeneity and not heterogeneity. Researchers contended that group stereotypes were supported because members of a minority group could be characterized with the group members who performed the poorest (Banks, 1993; Kitano & Kirby, 1986; Tonemah, 1987).

One way of defining giftedness concentrates upon those individuals with superior ability and potential; another way of describing giftedness emphasizes productivity and creativity (Gallagher, 1994). Gifted children, as defined by the first description, are early developers in walking, talking, and reading. They read two grade levels or more above grade level or have mastered advanced mathematical concepts and tasks prior to their peers. The second definition describes students who demonstrate productivity and creativity as valuable tools for society and self.

However, culturally different students are not thought of as gifted in either of these two ways (Colangelo & Davis, 1997). Each group of culturally different students has its own set of unique characteristics identified with the gifted. After teaching African-American students for 20 years and gifted students for 3 years, Gay (1978) developed a list of characteristics of African-American gifted students. The following five traits are among Gay's list of 13 characteristics in identifying African-American gifted students as different from the characteristics of gifted White students:

1. African-American gifted students respond more sensitively and quickly to racist attitudes and practices.

2. At a very early age, African-American gifted students may feel alienated by school.

3. Gifted African-American children have large vocabularies inappropriate for school. For the gifted African American, thinking in African-American English may hinder the facility of expression in standard English.

4. It is difficult to determine many areas of experiential knowledge for gifted African-American students.

5. Gifted African-American students have developed a strong ability for concentration due to persistent noise in their environment.

Gifted African Americans may not demonstrate their creativity in the same fashion as gifted White students. Using data generated by 36 teachers, who made observations on original behavior related to classwork, art, and antisocial behavior, Swenson (1978) developed a list of 25 creativity characteristics for culturally different and socioeconomically deprived urban-area students. Among these characteristics was that gifted African-American students might repeat activities so that they could do them differently. Gifted African-American students might use free time creating their own games or making something from paper and material scraps in contrast to more structured activities. Building and constructing things and using unusual materials or using ordinary materials in different ways were examples of creativity among the gifted African-American students. The gifted African American might think of his/her own ideas when the class was working on a project together. The gifted African American could find new ways to get attention. The gifted African American tried original ways to avoid work that he or she did not want to do. In wanting to know something, the gifted African American took the initiative and read about it or asked questions without prompting the teacher (Swenson, 1978).

Another important characteristic of gifted African Americans to add to the list was the strong and positive relationships that they usually had with their peers (Garmezy, 1991). Furthermore, Shade's (1978) summary of African-American achievers revealed that they were goal oriented, possessed great self-confidence, felt positive about themselves, felt in control of their destiny, had high levels of aspirations, and possessed confidence in themselves to be able to accomplish their goals (Maker, 1989).

Factors relating to success for gifted African-American students, according to Hauser, Vieyra, Jacobson, and Wertlieb (1989), included internal locus of control, self-confidence, and feelings of empowerment. Taylor (1991) identified maturity and academic and social confidence as characteristics of the gifted African-American student. Clark (1991) and Ford (1993) thought that the resilient, gifted African-American students had bicultural identities and that they believed in the American Dream. Positive school experiences, strong family values, and actively attending church and church-sponsored activities add to the description of gifted African-American students (Taylor, 1991; McLoyd, 1990).

These gifted and creativity characteristics of the African-American student can help teachers nominate and identify potential gifted African-American students. First, teachers need to be educated on the differences between the African-American students and White students. Then, caring, sensitive, and knowledgeable teachers of gifted students can help with this identification process among minorities.

The identification phase in most school districts includes intelligence tests, achievement tests, creativity tests, teacher nominations, parent nominations, and peer nominations (Davis & Rimm, 1998). Several improvements in this identification phase has occurred; for example, the Baldwin Identification Matrix (Baldwin, 1985) was developed to combine objective and subjective criteria. In using this matrix, Dabney (1983), Long (1981), and McBeath, Blackshear, and Smart (1981) reported that this matrix helped increase the number of African-American students identified for gifted programs.

Another successful identification model is the Frasier Talent Assessment Profile. This model includes aptitude and achievement test data in addition to self-report and observational information in data categories such as intelligence, special academic talents, motivations, creativity, and others.

Giftedness should be based on potential rather than actual academic performance. A low IQ or achievement score is not uncommon among the gifted disadvantaged, and IQ tests are considered racially biased (Frasier, Garcia, & Passow, 1995). Two creativity tests, the Torrance Tests of Creative Thinking and the GIFT and GIFFI inventories, are good instruments for finding creative gifted African-American students.

Giftedness in Mathematics

Piirto (1994) portrayed mathematicians as people who chose to spend their lives solving mathematical problems. These individuals differed from scientists in that they considered the process of solving the problem as important as obtaining the product. The mathematician and philosopher Bertrand Russell, at a meeting in Paris, discussed each evening for hours with Alfred Whitehead new mathematical and logical principles.

Professor Jerry P. King of Lehigh University in 1992 was quoted as saying that mathematics "is the loveliest subject on the face of the earth" (Piirto, 1994, p. 355). In Gustin's 1985 study of 19 male and 2 female exceptional research mathematicians, it was observed that these individuals worked hard and long hours and that they were "nearly obsessive" and that "their motivation became increasingly central" (Piirto, p. 355). These persons possessed the need to create, as was evident in their writings (Piirto, 1994).

To enable gifted mathematics students to realize their potential, goals for these very students were advocated by VanTassel-Baska (1994). These goals were:

1. To provide a context for gifted students to learn as much as possible about mathematical concepts, ideas, and skills. Since gifted students have the capacity to learn more than is usually presented in standard courses, adjustments in curriculum must be made.

2. To prepare mathematically gifted students to be creative and independent thinkers. Mathematics provides an environment for stimulating creative thought and developing the high potential of these students to become problem solvers of the highest caliber.

3. To help mathematically gifted students appreciate the beauty of mathematics. Through its study, gifted learners are more likely than other students to comprehend, value, and find meaning in mathematics as the study of patterns and as the language of the universe. (p. 232)

VanTassel-Baska (1994) claimed the content model to organize mathematics cannot be presented only through a supplemental program or enrichment materials; the daily program for these children should be closely aligned with their abilities. Careful planning needs to be made for these students' mathematics curriculum. Acceleration is one modification of mathematics curriculum in which students are placed in the course for the next higher grade level. However, if the curriculum is not changed significantly in pace or depth, it will not meet the needs of high-ability students. Another alternative is compacting the curriculum. By compacting the course, more time is allowed for additional enrichment topics or to move on to the next year's work (VanTassel-Baska, 1994).

Mathematically gifted learners demonstrate certain characteristics; for example, "speed in reasoning and ability to comprehend generalizations" (VanTassel-Baska, 1994, pp. 233-234). Research has suggested that mathematically gifted students attain the formal operation stage of reasoning in mathematics earlier than other students. Therefore, acceleration in mathematics content provides for an alternative for mathematically precocious students.

The NCTM (1983) policy statement on vertical acceleration recommended:

that vertical acceleration be considered only for a limited number of highly talented and mathematically creative students whose interest and attitudes clearly indicate that they have the ability and perseverance to complete a carefully designed sequential curriculum. For all but this select group, a strong, expanded program emphasizing mathematics enrichment is preferable. (p. 234)

Another good example of organizing the mathematics curriculum is using the process-and-product model. This particular model uses not only mathematical skills, but also organizational and research skills. It emphasizes problem-finding, problem-solving, and product development (VanTassel-Baska, 1994).

The concept model to organize mathematics curriculum is the third model. It uses themes or key ideas as key organizers for areas of study. One advantage of this model is that it tends to emphasize abstract concepts and modes of thinking. It is interdisciplinary and helps students apprise the worth of mathematical ideas (VanTassel-Baska, 1994).

Students' needs require that characteristics of the gifted and talented student be explored. Characteristics that distinguish mathematically gifted students from nongifted students studying mathematics include "spontaneous formation of problems, flexibility in handling data, mental agility of fluency of ideas, data organization ability, originality of interpretation, ability to transfer ideas, and ability to generalize" (Greenes, 1981). Their needs demand a curriculum that is deeper, broader, and faster than what is taught to the regular students. There is a saying in education that what is good for the gifted is good for all students. Gifted education has led in curriculum development for individual needs for the past 15 years. VanTassel-Baska (1994) has reinforced this by stating that good curricular design and delivery for the top 5% of the population can only strengthen the regular curriculum: "The curricular work for the gifted can spearhead higher standards and more rigorous methodologies in addressing the needs of the rest of the student body" (VanTassel-Baska, 1994, p. 4). Differentiating for the mathematically gifted should be in assessments, curriculum materials, instructional materials, and grouping models. These opportunities should be made available to any student with an interest in taking advantage of any of them (Johnson, 2000).

Real-World Mathematics Curricula for Gifted Students

The National Research Council, the NCTM, and the American Association for the Advancement of Science conducted studies that uniformly concluded that educators must take new directions. Educators must raise expectations for achievement and increase the breadth of offerings to include estimation, chance, symmetry, and exploration of data. Furthermore, educators need to engage students, demonstrate connections, and reduce fragmentation (Steen, 1990).

For all students, the NCTM's Standards (2000) emphasized reasoning, real-world problem solving, communication, and connection, rather than computational skills. The Standards recognized that all children are not the same and emphasized that all students should have opportunities to study and learn mathematics. Furthermore, the Equity Principle in the Standards advocated that high expectations for mathematics learning be communicated "in words and deeds to all students" (p. 13). The Standards Equity Principle supported a superior mathematics program that would provide a responsive learning environment to students' prior knowledge, intellectual strengths, and personal interests. In addition, the Standards Equity Principle announced that some students (for example, non-native English-speaking students) may need further assistance to meet high expectations. Also, students with disabilities may need extra time to complete assignments or need additional resources to help them meet high expectations. For example, an additional resource may be after-school programs, peer mentoring, or cross-age tutoring.

On the other hand, students with special interests or exceptional talent in mathematics may need enrichment programs or additional resources to engage their interest and to challenge them. These students, too, must be nurtured and supported so that they have the opportunity to excel in the classroom. The Standards (NCTM, 2000) added that educational systems must take care of the special needs of some students without inhibiting the learning of others. Implicit in these statements of equity principle is that the special needs of both gifted and nongifted students are to be considered in any mathematics curriculum.

The Standards (NCTM, 2000) stressed important mathematics, which included allowing "students to see that mathematics has powerful uses in modeling and predicting real-world phenomena" (p. 16). Many of these areas that the Standards was advocating for all students, in particular real-world problems, have been emphasized traditionally for gifted students (Johnson, 2000).

Real-world mathematics curricula were evident in the Boston Public Garden as teachers from across the United States attended "Explorations: A Projects Approach to Teaching K-8 Mathematics," a workshop organized to help teachers in their teaching of mathematical and cooperative learning skills. A section titled "Mathematics for Special Needs Populations: The Gifted Student and the Student with Learning Difficulties" helped teachers with skills for identifying both gifted students and students with learning difficulties. In addition, it provided instructional and assessment techniques for meeting these students' needs.

Teachers were led from the Remote Sensing Laboratory to the College of Engineering. They also ventured throughout the city of Boston from the new Star Market to the Make Way for Ducklings sculpture in the Public Garden. Teachers were to use the principles and theories that they had been learning in the workshop to develop real-world mathematics projects.

Teachers worked on small projects at Star Market. One of these projects was "In the Bag," in which middle school students packed a grocery bag with the goal of achieving the maximum number of items and the maximum cost while having the lightest weight. At the end of the project, students demonstrated selections and their calculations to the rest of the class. This exercise taught not only to estimate cost and determine weight and size, but also spatial visualization, within the framework of cooperative learning. "The kids don't always think of learning mathematics as grind, grind, grind. It makes it a lot more interesting for the teachers, too," said one participant. A workshop leader emphasized the importance of a real-world situation making connections for students.

At the Make Way for Ducklings sculpture, a project involving proportions and ratios took place. The history of the book, Make Way for Ducklings, was explained, and the location of author-illustrator Robert McCloskey's apartment, within sight of the pond, was pointed out. Workshop participants were told how McClosky had even filled his bathtub with ducklings in order to draw them accurately. This interdisciplinary approach to teaching created a real-world context for student learning. At the end of the 3-week workshop, a workshop leader presented the conclusion that students have to know how to add, subtract, and multiply, but the focus is now on knowing when to add, subtract, and multiply (Reilly, 1996).

With gifted students in mathematics, using the process-and-product model to organize mathematics curricula includes not only mathematical processes, but others such as organizational and research skills. This model's goal emphasizes problem-finding, problem-solving, and product development. Sternberg (1982) advocated problem-finding as a characteristic that distinguishes gifted scientists who make significant contributions to their fields. In this model, student interest influences project choice and direction. An example of a model curriculum of the process-and-product type is Challenge of the Unknown, a video-based problem-solving curriculum that features examples of real-world problem solvers. Many of these problem solvers are scientists or those who use the skills commonly related to scientific endeavors. Examples of extension projects are the study of fractals, design and implementation of a school census as a model of a national census, and the study of vectors in relation to a building demolition problem (VanTassel-Baska, 1994). Gallagher and Gallagher (1994) furthermore agreed with VanTassel-Baska (1994) by stressing utility and pertinence in the mathematics curriculum by applying it to different disciplines. The applications of mathematics in art are numerous, especially in geometry. Students can learn to apply geometric principles to create their own artwork. In addition, students can engage in optical art, studies in dimension, and historical use of patterns in pottery and engravings as they systematically use different geometric shapes and forms in geometry. Opportunities for mathematics and social science are numerous. For example, the representation and interpretation of statistics can pique the curiosity of gifted students through interpreting two graphs in which numbers can be misconstrued to represent findings. The calculator is used with gifted students to illustrate the relationship between algebraic and spatial representations of different mathematical formulas.

The National Science Foundation funded the Core-Plus Mathematics Project (CPMP), a curriculum for the gifted and all students. The CPMP was tested in schools across the country with diverse student populations, including mathematically promising students. The curriculum was planned to advance students' mathematical thinking each year along four interwoven strands: (a) algebra and functions, (b) geometry and trigonometry, (c) statistics and probability, and (d) discrete mathematics. The CPMP curriculum was designed around a coherent set of important, broadly useful mathematical concepts and methods. Topics for the gifted included networks, Monte Carlo methods, and computer and calculator graphics (Hirsch & Weinhold, 1999). When school mathematics is seen as much more than arithmetic, generalized arithmetic, precalculus, and calculus, students with interests and ability in the mathematics of data analysis, shape, discrete structures, or chance also have opportunities to excel.

By taking advantage of workshops, such as the Boston Garden project and similar mathematics programs around the country, teachers return to their classroom in the fall prepared to provide a real-world mathematical opportunity for learning that accommodates the needs of all students in the mathematics classroom. The teachers participating in the workshop represented a broad spectrum of teaching specialties, including learning disabled, culturally diverse, and gifted. Clearly, the instructor envisioned (and passed this vision to the workshop participants) that a real-world mathematical curriculum is successful with all students: gifted, nongifted, and special needs.