Mathematics Reform in the United States
The need to improve student mathematical achievement is a pervasive issue in the United States. This issue has received greater interest in recent years because of the United States' mediocre academic performance in comparison to their international counterparts (A+ Education Foundation, 2005). Mathematics competence opens doors for those students, but those same doors will be closed to those who lack competence (NCTM, 2000). Both NCTM and National Research Council (NRC) agree that the central tenet of mathematics education is that, in order to do mathematics well, students should learn important mathematics concepts and processes with understanding. Achieving this goal requires solid mathematics curricula as well as competent and knowledgeable teachers who are committed to equity and excellence. Students need and deserve the best mathematics education possible to enable them to fulfill their career goals in an ever-changing world (NCTM, 2000).
This paper begins with a historical review of mathematics reform in the United States, then discusses the need for a national standards and curriculum for mathematics and ends by discussing three programs developed with the support of the National Science Foundation and which of these programs should be the national curriculum in the United States.
For many students in the United States, mathematics consists of memorizing facts and procedures. Many educators have maintained that if students are able to perform a set of procedures is then they know the mathematics (Battista, 1999). Even though some topics are taught year after year, students are not learning them. This means that traditional methods of teaching mathematics are ineffective. The National research Council has reported that 60% of college students are enrolled in mathematics that is ordinarily taught in high school. Many adults readily admit to experiencing difficulty with the simplest math skills (Battista, 1999). The level of mathematical thinking and problem solving needed in the 21st Century workplace has increased dramatically. Students need to be prepared for solving problems in a variety of school, home and work settings (National Council of the Teachers of Mathematics, 2000).
Evidence of problems with education in the United States was confirmed when the Third International Mathematics and Science Study (TIMMS) was conducted worldwide in 1995. Findings from these studies prompted the National Council of the Teacher of Mathematics (NCTM) to issue the Principles and Standards for School Mathematics (PSSM) in 2000. PSSM is a vision statement for mathematics education in the United States and calls for the development of a core curriculum in mathematics that prepares all students for life after high school. This document gives educators a clear picture of what students are expected to know and how teachers should foster that learning. The PSSM places great emphasis on problem solving, reasoning and proof, and the ability to communication of mathematics (NCTM, 2000).
While the NCTM Standards were not mandated, it was hoped that states would adopt them (Reys, Robinson, Sconiers, & Mark, 1999). The teacher in a standards-based classroom selects good tasks, engages students in thoughtful reflection, and creates a classroom environment that supports communication. Students are physically and intellectually engaged in real-world problems and tasks and are working independently or within a group (NCTM, 2000).
For students to become mathematical thinkers and problem solvers there must be implementation of a conceptual, problem-solving approach to mathematics instruction. This means teachers have to have a deep understanding of both mathematics and mathematics pedagogy. Conceptual understanding enables students to deal with any situation they encounter (Gearhart, Saye, Seltzer, Schlackman, Ching, Nasir, Fall, Bennet, Rhine, & Sloan, 1999). However, teachers must proper materials and professional development to ensure they are prepared to standards based teaching (Briars, 1999).
Having realized the importance of proper materials, by 1996 many states were following NCTM content standards recommendations. In addition, the National Science Foundation (NSF) had funded a series of comprehensive curriculum development projects that resulted in the development of "standards-based" curricula for elementary, middle and high schools (Martin & Berk, 2001).
National Standards and Curriculum
On September 21, 2009 the National Governors Association Center for Best Practices (NGA Center) and the Council of Chief State School Officers (CCSSO) released a draft copy of the College and Career Readiness Standards for Mathematics. This document was released by the NGA Center and the CCSSO in partnership with Achieve, ACT and the College Board. Governors and state commissioners of education from across the country committed to joining a state-led process to develop a common core of state standards in English-language arts and mathematics for grades K-12 (Common Core Standards Initiative, 2009).
The Common Core State Standards Initiative builds on years of standards efforts led by states and national organizations. The first step of this initiative is developing college and career-readiness standards followed by K-12 standards. These standards will be research and evidence-based, internationally benchmarked, aligned with college and work expectations and include rigorous content and skills (Common Core Standards Initiative, 2009).
The NGA Center and CCSSO are coordinating the process to develop these standards and have created an expert validation committee to provide an independent review of the common core state standards, as well as the grade-by-grade standards. The College and Career Readiness Standards for Mathematics is divided into three parts: a Standard for Mathematical Practice; ten Standards for Mathematical Content; and a set of Example Tasks. The Standard for Mathematical Practice outlines the core practices that are essential for students being proficient for college and the workforce. According to the Standards proficient students attend to precision, look for structure, note regularity and use technology intelligently. The Mathematical Content Standards list ten core concepts, core skills and a description of each to pull together previously taught topics and deepen understanding for future topics. These content standards are: number, quantity, expressions, equations, functions, modeling, shape, coordinates, probability, and statistics. The third component of the College and Career Readiness Standards for Mathematics are a collection of web based tasks that allow students to use the knowledge they have been taught (Common Core Standards Initiative, 2009).
The primary goal of these standards is to enable students to achieve mathematical proficiency. Instruction, curriculum and assessment should be designed to achieve these standards across multiple levels of educational systems: state, district, school and teacher. It is the intention that the implementation of the College and career Readiness Standards for Mathematics will eliminate the need for remedial college mathematics and prepare students for training programs for career level jobs (Common Core Standards Initiative, 2009).
National Science Foundation and Mathematics Curricula
Since its beginning in 1950, the National Science Foundation (NSF) has played a strong, role in making American scientific research and technological application the best in the world (McKeown, Klein, Patterson, 2002). The purpose of the NSF is to promote the progress of science; to advance the national health, prosperity, and welfare; and to secure the national defense. With federal funding from the National Science Foundation, over a dozen sets of new K-12 mathematics curriculum materials were developed in the 1990's to support the recommendations of the NCTM Standards. A significant number of research studies have investigated student learning with standards-based materials (Huntley, Rasmussen, Villarubi, Sangtong, & Fey, 2000; National Research Council, 2004; Schoen, Cebulla, Finn, & Fi, 2003; Senk & Thompson, 2003).
In addition, a growing field of research has focused on teachers' experiences using and learning from standards-based curriculum materials (Arbaugh, Lannin, Jones, & Park-Rogers, 2006; Clarke, 1997, 1999; Collopy, 2003; Frykholm, 2004; Lloyd, 1999; Manouchehri & Goodman, 1998; Remillard, 1999, 2000; Remillard & Bryans, 2004; Van Zoest & Bohl, 2002; Wilson & Goldenberg, 1998).
Although these reports have offered encouraging images of teachers' abilities to learn new subject matter and develop new pedagogical practices through the implementation of supportive curriculum materials, they have also illustrated the complexity of the challenges teachers face as they use new materials for mathematics instruction. It is important now for researchers to identify and explore specific obstacles and challenges encountered by teachers when they attempt to teach mathematics in unfamiliar ways.
Core-Plus is a National Science Foundation-funded project that, during the mid-1990s, developed and field-tested student and teacher materials for a 4-year high school mathematics curriculum. As elaborated by its designers (Hirsch, Coxford, Fey, & Schoen, 1995; Schoen & Hirsch, 2003), the curriculum aims to support teachers in enacting many components of the recommendations of the standards (NCTM, 1989).
The Core-Plus materials recommend the use of multiple instructional formats, including whole-class discussions for introductions and summaries of activities, cooperative groups for class work, and individual work for homework. To accompany the new multiple instructional formats are multiple roles for the teacher and students. In the whole-class formats, the Core-Plus teacher directs, encourages and coordinates students' involvement in class discussions while also providing guidance and focus about the particular investigation, concept, or question. The envisioned cooperative group format demands that the teacher acts as a facilitator to promote and support the students' active explorations of the mathematical situations presented in the Core-Plus materials. Within each of these roles, the teacher-student interactions should contain discourse in which the teacher solicits ideas from students and provides guidance towards the development of important mathematical ideas.
Core-Plus is a text series cited as one of the worst reform mathematics texts by groups such as Mathematically Correct. They cite schools that have dropped Core-Plus after adopting it, and cite studies showing graduates of the curriculum scoring poorly in college math placement exams (Mathematically Correct Website, 2009). Other groups, such as Mathematically Sane, have applauded the NCTM-Standards-based curricula such as Core-Plus Mathematics. They point to the many schools in which Core-Plus is successfully used, and cite research studies supporting the effectiveness of Core-Plus Mathematics that have appeared in journal articles, book chapters, conference presentations, and Ph.D. dissertations (Mathematically Sane Website, 2009).
Interactive Mathematics Program
The Interactive Mathematics Program (IMP) is a growing collaboration of mathematicians, teacher-educators, and teachers who have been working together since 1989 on both curriculum development and professional development for teachers. With the support of the National Science Foundation, IMP has created a four-year program of problem-based mathematics that replaces the traditional Algebra I-Geometry-Algebra II/Trigonometry-Pre-Calculus sequence and that is designed to exemplify the curriculum reform called for in the Curriculum and Evaluation Standards of the NCTM.
The IMP curriculum integrates traditional material with additional topics recommended by the NCTM Standards, such as statistics, probability, curve fitting, and matrix algebra. IMP units are generally structured around a complex central problem. Although each unit has a specific mathematical focus, other topics are brought in as needed to solve the central problem, rather than narrowly restricting the mathematical content. Ideas that are developed in one unit are usually revisited and deepened in one or more later units.
IMP is among the reform curricula that have been heavily criticized by organizations such as Mathematically Correct. That organization's Internet site begins with a statement that "advocates of the new, fuzzy math" (focus) "on things like calculators, blocks, guesswork, and group activities and they shun things like algorithms and repeated practice. The new programs are shy on fundamentals and they also lack the mathematical depth and rigor that promotes greater achievement (Mathematically Correct Website, 2009).
Supporters point to statistical studies that compare the performance of students enrolled in IMP courses with their peers enrolled in traditional high school mathematics courses. Merlino and Wolff, two such researchers, report that in their several studies IMP students consistently outperformed traditionally taught students on both the math and verbal sections of the PSAT, as well as on the SAT-9 (Merlino and Wolff, 2001).
The IMP curriculum has been thoroughly field-tested and enthusiastically received by hundreds of classroom teachers around the country. Their enthusiasm is based on the success they have seen in their own classrooms with their own students.
These informal observations are backed by more formal evaluations.
Dr. Norman Webb, of the Wisconsin Center for Education Research, has done several studies comparing the performance of students using the IMP curriculum with the performance of students in traditional programs. For instance, Dr. Webb has found that IMP students do as well as students in traditional mathematics classes on standardized tests such as the SAT. This is especially significant because IMP students spend about 25 percent of their time studying topics that are not covered on these tests. To measure IMP students' achievement in these other areas, Dr. Webb conducted three separate studies involving students at different grade levels and in different locations. The three tests used in these studies involved statistics, quantitative reasoning, and general problem solving. In all three cases, the IMP students outperformed their counterparts in traditional programs by a statistically significant margin, even though the two groups began with equivalent scores on eighth-grade standardized tests.
The development philosophy of MathScape centers on the belief that mathematics is linked to the human experience. People use mathematics to explore their surroundings, build structures and communities, and to seek to understand the world around them. The MathScape curriculum provides meaningful mathematical investigations that allow students to experience these different uses of mathematics and, in so doing, to build mathematical competence. The MathScape curriculum reflects the following principles about mathematics: (a) mathematics is a way of thinking; (b) mathematics is connected to other subjects both in and out of school; and (c) topics within mathematics are connected to one another.
A good mathematician can fuse pieces of content knowledge in creative ways to arrive at mathematical results. Mathematicians develop habits of mind such as abstracting essential elements from situations and seeking patterns and relations; mathematicians who have developed habits of mind have an intuitive sense for how to approach new problems. MathScape activities aim to develop productive habits of mind in students; all activities are designed to get students to think critically and creatively about mathematical topics.
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