*Douglas A Grouws & Melissa D McNaught. 21st Century Education: A Reference Handbook. Editor: Thomas L Good. Sage Publication. 2008.*

Difficult mathematics classes were once considered necessary to filter out untalented students to ensure that the country had the skilled scientists and engineers needed to promote the nation’s economic progress in a context of rapid social change and technological advancement. But today a strong mathematics education is considered a necessity for all students to be successful in their personal lives and in the workplace (National Council of Teachers of Mathematics [NCTM], 2000). Now, mathematics courses must function as a “pump instead of a filter” (White, 1988), moving all students successfully through the system. When considering how to reach the goal of “mathematics for all,” one must consider every facet of mathematics education. The field is deeply rooted in mathematics, but must also take into account findings from other disciplines such as psychology and sociology. Classrooms involve complex dynamic systems, and many factors that are often considered the purview of other disciplines play vital roles in the amount and nature of the student mathematics learning that occurs. Despite this complexity, three constructs are important in any mathematics classroom: curriculum, teaching, and assessment.

The mathematics curriculum determines what content students study, while teaching involves interactions that shape the quality and the depth of the mathematics learned. Assessment provides important feedback for structuring and moving the learning process forward. Thus, attaining significant mathematics achievement “for all” requires a coherent curriculum, knowledgeable teachers employing effective teaching practices, and the use of assessments that accurately measure what students learn. This chapter addresses curriculum, teaching, and assessment with the realization that the interplay among these factors and out-of-class factors such as socioeconomic status are as important as the factors themselves.

## Curriculum

Curriculum can be interpreted in various ways. Generally curriculum refers to the content students are expected to learn. However, when individuals speak of curriculum, they may be referring to instructional materials, textbooks, state standards, grade-level expectations, or lesson plans. Regardless of how one defines curriculum, the reality is that all of these determine what mathematics students are given the opportunity to learn.

The mathematics content students have been expected to learn has varied over time in response to many factors. For example, in 1957, the Soviet Union launched the Sputnik I satellite forcing the United States to reconsider its perceived position as the leader in space technology. After Sputnik’s launch and with a concern for national security, President Eisenhower formally introduced the Space Race and teachers soon found themselves expected to prepare a new generation of engineers, scientists, and technologists. Consequently, during the 1960s, new mathematics curricula were designed and produced for this purpose. The term New Math refers to these curricular programs and they initiated dramatic changes in what mathematics was taught and how it was taught.

Directed and influenced by academic mathematicians, the driving force underlying these curricula was the idea that certain concepts, structures, and reasoning processes provided a common foundation for the specific topics taught within mathematics. A central tenet was that concentrating on unifying concepts such as sets, relations, and functions would allow students to develop a deeper understanding of mathematics rather than a focus on disconnected procedures learned through rote training. Developers of New Math soon discovered that writing new textbook materials was not sufficient to promote their intended changes. Teachers needed training to be able to implement the new textbooks and administrators needed to communicate to their communities the rationale behind reforming the mathematics curriculum. The poor student performance that followed these curricular changes was blamed on the New Math and the criticism is just, at least in part, but the mathematicians who created it continued to blame the lack of proper curricular implementation by the teacher in the classroom as the true cause of its demise. Parents became concerned about their children’s mathematical learning and looked to the media to help convey their plea to go “back to the basics.”

With the public perception that New Math materials were not satisfactory, the 1970s began to emphasize the development of basic computational skills and efficient algebraic manipulation. These curricular materials abandoned the formal language and concepts associated with the New Math materials and returned to an emphasis on more familiar ideas of computation. The National Science Foundation funded several conferences with the intention of improving mathematics education. Emerging from these conferences were concerns expressed by mathematics educators that skills were being emphasized without regard to their application or to the process of problem solving. Arguments about what mathematics is “basic” were heated and frequent following these conferences. This dialogue eventually led the National Council of Teachers of Mathematics (NCTM) to issue their Agenda for Action (1980) report. This agenda redefined basic skills to encompass much more than computational fluency. NCTM recommended that problem solving become the focus of school mathematics in the 1980s.

In 1983, the U.S. Secretary of Education released A Nation at Risk, a report that in large part proclaimed the demise of public education. “Our society and its educational institutions seem to have lost sight of the basic purposes of schooling, and of the high expectations and disciplined effort needed to attain them” (National Commission on Excellence in Education, 1983, p. 7). This document called for reform efforts in which schools were to “demand the best effort and performance for all students, whether they [were] gifted or less able, affluent or disadvantaged, whether destined for college, the farm, or industry” (p. 24). Since then vast resources have been spent on mathematics reform “for all.”

Subsequently, NCTM published its vision of what school mathematics should be under the title Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989). This vision represented a consensus that “all students need to learn more, and often different, mathematics and that instruction in mathematics must be significantly revised” (p. 1). This document quickly gained acceptance and was later widely endorsed by legislators and professional organizations such as the American Mathematical Society (AMS) and the Mathematics Association of America (MAA). With these endorsements the NCTM’s Curriculum and Evaluation Standards began to reach the classroom.

The Curriculum and Evaluation Standards challenged the fundamental beliefs that had been held by many mathematicians, mathematics educators, and the general public for a number of years. It outlined an active learning environment (as opposed to the long-established focus on the processes of memorization and practice) and a greater emphasis on developing meaning and building conceptual understanding. With each grade level band (K-4, 5-8, 9-12), the Standards recommended concepts and skills that should receive increased attention, for example, real-world problems and statistics, and those that should receive decreased attention, e.g., complex paper-and-pencil computation. The Standards contended that all students could succeed with higher-level mathematics, if engaged in an investigative environment emphasizing conceptual understanding and collaboration among peers.

The de-emphasis on computational skills in the Standards soon came under attack and was often inappropriately publicized as the elimination of such skills from the curriculum. With the driving force of the Standards being mathematics for all, conservative mathematicians and some mathematics educators feared that the mathematics curricula would become “dumbed-down” in order for more students to be successful. Nevertheless, the National Science Foundation (NSF) called for and funded proposals for the development of curricula that would align with the Standards. The fundamental differences in beliefs and the discussions about what and how mathematics should be taught is what is often referred to by the term math wars (see Schoenfeld, 2004).

The shift away from memorization and rote application of procedures required that new materials be developed to help teachers promote students’ conceptual understanding of mathematics. Historically, mathematics textbooks in the United States have integrated mathematics content at each elementary school grade level, while at the high school level they have traditionally segregated content according to specific foci such as algebra, geometry, and precalculus. When, in 1990, the National Science Foundation (NSF) began to finance the development of new curricula to embody the new vision set forth by the Standards the new textbooks that were developed were dubbed “standards based” materials, because they were meant to embody the instructional recommendations outlined in the Standards. Written primarily by mathematics educators with strong mathematical backgrounds, these materials were quite different from previous textbooks. The situations and problems posed often involved multiple possible solutions and solution strategies. They did not focus on repeated practice on given problem solving methods on strings of similar problems. The teaching method embodied in these materials focused on small-group work with increased attention to the use of technology, as will be discussed later.

In elementary and middle schools, textbook lessons were set in real-world contexts and were more problem-based than previously published textbooks. There was less emphasis on practice with computational algorithms, and in some programs there was a tendency to move content to lower grade levels. At the high school level, the conventional single-subject textbooks were replaced by textbooks that integrated the study of geometry, algebra, statistics, and so on, at every grade level. Thus, the high school materials formed an integrated approach to learning mathematics by including content from each content strand in each grade level textbook. The idea was that this would promote forming connections across mathematical ideas and develop a more coherent view of mathematics in general. These curricular materials are now typically called “integrated” mathematics. For more information regarding these “standards-based” curricular materials, see www.comap.com/elementary/projects/arc(Grades K-4), showmecenter.missouri.edu (Grades 5-8), or www.ithaca.edu/compass (Grades 9-12).

As mathematical reform swept the nation, the label of “standards-based curriculum” became prevalent. Curriculum once referred to as traditional even began to claim the label standards-based, “[y]et there is a significant difference between texts that have retrofitted their traditional ‘demonstration and practice’ approaches in order to better align themselves with the NCTM Standards, and curricula that were designed from the outset to embody the mathematical approaches and pedagogical principles advanced by the Standards” (Goldsmith, Mark, & Kantrov, 1998, p. 10). Few publishers made significant changes to integrate mathematical topics in the way the Standards recommended (Taylor & Tarr, 2003) perhaps reflecting a marketing concern that major changes would not be well received.

Critics of the new materials emerged and are becoming increasingly vocal. They call the curricula developed during this reform movement as “the new New Math” or “fuzzy math” (Grouws & Cebulla, 2000). Arguments reminiscent of the 1960s New Math era have been reincarnated as parents have become concerned that the mathematics their students need to know, especially basic skills, might not be learned. The recommendation to reduce emphasis on memorizing and practicing procedures has often been misinterpreted to mean that learning procedures are not necessary.

In 2000, the Standards were updated, refined, and published under the title, Principles and Standards for School Mathematics (PSSM). The revision took account of recent research (Kilpatrick, Martin, & Schifter, 2003), current thinking about what is most important to include in school mathematics (e.g., more attention to data analysis and informal statistics), and changes in mathematics (e.g., a focus on representation). The document also addressed some criticisms of the earlier Standards document. PSSM placed a greater emphasis on the importance of algorithms and computational fluency than the previous document. Its publication has not stemmed the discussion of what mathematics should be studied and how it should be taught.

__Influences on Curriculum__

Many influences shape the school curriculum, and thus the mathematics that students have an opportunity to learn. These factors include textbooks, state and local learning expectations, and mandated assessments. Furthermore, the emphasis placed on specific mathematical topics is affected by individual teachers’ mathematical knowledge, instructional beliefs, and priorities. In recent years, the federal No Child Left Behind legislation (NCLB, 2002) mandated that states assist schools in closing the achievement gaps that exist among different groups of students (e.g., racial groups) by adopting and specifying challenging academic content that would be used by all districts in the state. In response, states developed new specific objectives which have become known as the states’ grade-level expectations (GLEs). These expectations describe in detail what mathematics students are expected to learn at each grade level. Publishers utilize these GLEs in making decisions regarding what mathematics content to include in their textbooks. Developing textbooks that align to multiple state documents is difficult, particularly because there is little consensus among the states regarding the grade placement of mathematical topics (see Reys, 2006).

Dissatisfaction with student performance on international tests and the implications of these results have prompted further discussion about curriculum issues and actions to communicate more clearly what is important mathematics to teach. One example is NCTM’s Curriculum Focal Points for Prekindergarten Through Grade 8 Mathematics: A Quest for Coherence (2007). This document identifies three essential mathematical topics to be comprehensively taught at each grade level (K-8) and its intent was to bring more curricular coherence across the nation’s classrooms. The document provides a beginning point for states and districts to design more focused curricular expectations to address the common criticism that mathematics curriculum in the United States is “a mile wide and an inch deep” (Schmidt, McKnight, & Raizen, 1997). Other organizations that have not been satisfied with NCTM’s efforts have developed their own recommendations to address their perception of the educational shortcomings that have resulted in low performance on international comparisons. See for example, the mathematics standards published by: Achieve ( http://achieve.org/node/479), American Statistical Association (www.amstat.org/education/gaise), and College Board (www.collegeboard.com/about/association/academic/standard.html).

Regardless of the standards in place, or the curriculum in use, curriculum alone is insufficient to determine student learning. Teaching also plays a major role in shaping students’ learning opportunities. The emphasis teachers place on different learning goals and topics, the expectations for learning that they set, the time they allocate for particular topics, the kinds of tasks they pose, the kinds of questions they ask, the types of responses they accept, the nature of the discussions they lead—are all parts of teaching and heavily influence what students learn and how they learn it.

## Teaching

Surprisingly, even as late as the 1970s, the case needed to be made that teachers do, in fact, make a difference in what students learn in the classroom. As research progressed, it became exceedingly clear that teachers do indeed make an important difference (see for example, Good, Biddle, & Brophy, 1975). Important to note, some teachers are particularly effective in promoting student learning year in and year out with different groups of students. Even more important, it has become clear that it is what teachers do when they teach that seems to be particularly important, as opposed to what teachers are. That is to say that variables such as teaching experience, degrees held, content courses taken, and so on are not as important as what teachers actually do in the classroom. In other words, teaching is not the same asteachers. The focus in this section is on teaching—classroom interactions among teachers and students around content directed toward facilitating students’ achievement of learning goals. Characteristics of teachers can certainly influence their teaching, but these characteristics do not determine their teaching.

Documenting what particular features of teaching directly influence student learning is different. Classrooms are a complex system where teaching is embedded in a system of many different interacting features, thus the effect of teaching on student learning cannot be measured independently of the system in which they operate. It is this system that affects student learning, not the individual features of a particular teacher or teaching method. Furthermore, the results of teaching are mediated by the student (e.g., their interpretation of instruction, their time on task, and their prior knowledge).

It seems reasonable to tailor teaching methods to what we know from research rather than from intuitive notions that might be true or false, or from scholars’ armchair speculations. The key to success in indetifying useful research findings is to look for patterns across teaching studies that produce similar positive effects on student learning within a common teaching goal. Following Hiebert and Grouws (2007), attention is now given to particular patterns that emerge from such an analysis in two goal areas: teaching for skill efficiency and teaching for conceptual understanding.

Different kinds of teaching facilitate different kinds of learning based on what students have had the opportunity to learn. Interestingly, there is not a simple correspondence between one method of teaching and skill efficiency and between another method of teaching and conceptual understanding. The best way to express current knowledge in the field is to describe some features of teaching that facilitate skill efficiency and some features of teaching that facilitate conceptual understanding, and to indicate where these features intersect.

__Teaching for Skill Efficiency__

Skill efficiency is defined as the rapid, smooth, and accurate execution of mathematical procedures. Results from several studies indicate that effective teaching toward this goal is characterized by a rapid instructional pace and includes extensive teacher modeling of the procedures to be learned with many teacher-directed product-type questions (see Good, Grouws, & Ebmeier, 1983). The teaching also displays a smooth transition from teacher demonstration to substantial amounts of error-free practice by students. Noteworthy in this set of features is the central role played by the teacher in organizing, pacing, and presenting information to meet well-defined learning goals.

The features of teaching identified above facilitate students’ skill efficiency but what effect these features have on conceptual understanding is less clear because they have not been thoroughly researched.

__Teaching for Conceptual Understanding__

Conceptual understanding can be defined as the construction of relationships among mathematical facts, procedures, and ideas (Brownell, 1935), and its positive effects on student learning have been well demonstrated in research programs as The Missouri Mathematics Project (Good, Grouws, & Ebmeier, 1983). Two features of instruction emerge from the literature as especially likely to help students develop conceptual understanding of the mathematics topic they are studying: (1) attending explicitly to concepts; and (2) encouraging students to wrestle with the important mathematical ideas in an intentional and conscious way.

Attending explicitly to concepts refers to treating mathematical connections among facts, procedures, and ideas clearly and meaningfully. This includes such things as discussing the mathematical meaning underlying procedures, asking questions about how different solution strategies are similar to and different from each other, considering the ways in which mathematical problems build on each other or are special (or general) cases of each other, attending to relationships among mathematical ideas, and reminding students about the main point of the lesson and how this point fits within the current sequence of lessons and ideas.

In many ways, the claim that students acquire conceptual understanding of mathematics when teaching attends explicitly to mathematical concepts is a restatement of the general observation that students learn best what they have an opportunity to learn. This claim has support from numerous research studies that span multiple contexts and designs (see Floden, 2002). Both teacher-centered and student-centered teaching methods that have explicitly attended to conceptual development have shown higher levels of students’ conceptual understanding than similar methods that have not attended to conceptual development directly. The ways in which concepts are developed in classrooms can vary—from teachers actively directing classroom activity to teachers taking less active roles, from methods of teaching that highlight special tasks or materials to those that highlight special forms of classroom discourse to those that highlight student invention of solution strategies. The evidence does not justify a “best” method of instruction to facilitate conceptual understanding, but it does support a feature of instruction that might be part of many methods: explicit attention to conceptual development of the mathematics. Students receiving such instruction develop conceptual understanding to a greater extent than students receiving instruction with less conceptual focus.

A second aspect of teaching associated with increased conceptual understanding is allowing students to struggle with important mathematical ideas. The word struggle is used to mean that students work at making sense of the mathematics; to figure out something that is not immediately apparent. This does not mean posing overly difficult problems that result in high levels of student anxiety and frustration. Rather, struggle here refers to solving problems that are within reach and grappling with key mathematical ideas that are comprehendible but not yet well formed (Hiebert et al., 1996). Struggling is the antithesis of being presented information to be memorized or being asked to practice what has been demonstrated by numerous examples. When students struggle, they devote high levels of energy to make sense of a situation rather than turning immediately to a prescribed and rehearsed method they have recently seen demonstrated. The struggle leads students to construct interpretations that connect new ideas to what they already know and to reexamine and restructure what they have already learned. This, in turn, yields content and skills that are learned more deeply and can be applied more easily to novel situations.

__Teaching with Technology__

One aspect of teaching that is gaining a lot of attention as an aid to increasing conceptual understanding is the use of various kinds of technology during instruction. Technology is engrained in our society and historically has yielded important scientific, economic, and cultural advances. However, the issue of technology in mathematics education has been a divisive topic for over two decades. On the one hand, many scholars feel that teachers should help students use technology to expand their knowledge beyond the limits of their skills to explore concepts that would be beyond their reach without technology. Other educators resist the notion of using technology for fear that students will become overly reliant on it and may even become unable to perform basic computations. In the 1989 Curriculum and Evaluation Standards, the National Council of Teachers of Mathematics called for all students to have access to calculators and computers at every grade level in order to investigate and solve problems. Having access to a tool does not guarantee its proper use. “One needs to be careful not to give the impression that technology itself makes the difference in teaching and learning. It is, of course, not the technology that makes the difference but rather how it is used and by whom” (Heid, 2005, p. 348).

Classroom technology can go well beyond the use of calculators. For example, virtual manipulatives are becoming more commonly used in the elementary grades. A virtual manipulative is defined as “an interactive, Web-based visual representation of a dynamic object that presents opportunities for constructing mathematical knowledge” (Moyer, Bolyard, & Spikell, 2002, p. 373). These are often replicas of frequently used concrete manipulatives such as pattern blocks, base-ten blocks, geometric solids, or colored rods. The use of these virtual manipulatives seems to promote making connections among various representations of concepts. They create a bridge between concrete 3-D manipulatives and the 2-D visual representations of the objects and eventually to the building of symbolic notations of abstract mathematical ideas. The flexible nature of the virtual pictorial representations, as opposed to static pictures, allows students to engage in investigation and clarification of their own thinking through experimentation (Moyer, Niezgoda, & Stanley, 2005). Many virtual manipulatives programs and applets are available free on the Internet. For more information visit the Web site of the National Library of Virtual Manipulatives ( http://nlvm.usu.edu/en/nav/vlibrary.html).

Research at higher grade levels has established how graphing calculators can be used as a tool for connecting mathematics to other disciplines (Garofalo, Bennett, & Mason, 1999). One skill, for example, that students need to develop in any discipline, and to a greater extent in real life, is the ability to interpret graphs and statistical data. The graphing calculator as part of class discussion around examples from economics, geography, and civics can be used for “simplifying data gathering [to allow] more time for analyzing and interpreting data” (Drier, Dawson, & Garofalo, 1999, p. 21). Students can graph data and use the information to discuss the effects of economy, climate, latitude, and energy resources on the data. Scales of axes can be changed easily to discuss visual bias and effective analysis. “These types of explorations help teachers prepare students to become logical thinkers who are able to apply mathematics in the real world” (Drier, Dawson, & Garofalo, 1999, p. 25). “These activities promote authentic learning in that students are manipulating, determining, interpreting, and analyzing information relevant to real-world situations” (Garofalo, Bennett, & Mason, 1999, p. 104). This process ultimately prepares students to draw inferences and make informed decisions. Other software programs such as Tinker Plots and Fathom have also been specifically designed for students to investigate and analyze statistical concepts (for more information, see www.keypress.com). Geometric software programs such as the Geometer’s Sketchpad (see www.keypress.com) have been found to be useful in helping student develop a comprehension of many geometric ideas and in fostering an understanding of graphical representations of algebraic concepts.

Computer algebra system (CAS) has become a common term used to refer to “tools that perform symbolic manipulation as well as generate graphs and perform numerical calculations” (Heid, 2005, p. 348). CAS has greater educational potential than a graphing calculator because it can manipulate algebraic expressions and equations in their symbolic form. These symbolic manipulations include simplifying algebraic expressions, changing forms of expressions, differentiation, factorization, solving equations, taking limits, series expansion, series summation, and matrix operations. This powerful tool can facilitate student mathematical investigations and discoveries (Drijvers, 2003).

The advantages of CAS are in the rapid execution of complex algorithms that typically bog students down when trying to understand a difficult concept. With a CAS system doing the procedural work, students can concentrate on understanding the concepts being considered and thus make sense of the mathematics. This results in a greater emphasis being placed on interpreting the meaning of results and determining when to apply information. Furthermore, “students are able to access high-level mathematical processes previously inaccessible to them” (Heid, 2003, p. 36). The role of CAS as an organizer for procedural tasks that must precede certain conceptual developments allows students to advance their conceptual mathematics knowledge before mastering complicated procedures. This gives a special advantage to a student who struggles with procedures so that he or she can still understand concepts. The higher level of conceptual knowledge that is garnered helps make the procedures involved make more sense.

Unfortunately, without proper facilitation by the teacher, students may become overreliant on the technology and generate answers they are unable to interpret. Thus, technology might be treated as a timesaving device with little emphasis on how to evaluate nonsensical answers and ignoring “algebraic insight” which can be derived from the procedural mathematics.

Technology is not a panacea, it can never replace effective teaching. It can, however, be a tool at the disposal of every teacher and student. Its effective use depends on skilled guidance by technologically knowledgeable teachers. The union of skilled teachers and intellectually engaged students generate the power of technology. Teachers must be mindful of the appropriate time to introduce technology in the classroom. Knowing when to use a tool is often as important as knowing how to use it. There should always be a basic understanding of the tool before its regular use in order to enhance learning. Clements and Sarama (2005) noted, “Computer programs should help and empower children to learn and meet specific educational and developmental goals more effectively and powerfully than they could without the technology” (p. 64)

When structuring one’s teaching based on research, one should first consider the learning goals for students. Research suggests different teaching strategies depending on whether one’s goal is skills development or conceptual development. A contrast has been made between teaching that encourages conceptual understanding and teaching that focuses on skill efficiency. Although these are not contradictory goals, they would likely be emphasized to different degrees in different systems of teaching, and the teaching features associated with reaching each goal are different. But it is important to keep in mind that studies show skill learning is relatively high in classrooms where features of teaching associated with conceptual development are being implemented.

## Assessment

Various types of mathematics assessments are employed at the classroom, state, national, and international levels. As districts struggle to meet the adequate yearly progress (AYP) required by the No Child Left Behind Act of 2001 (NCLB, 2002), the link between curriculum and assessment has come under increased scrutiny by teachers, administrators, researchers, and state education department personnel. Almost all states have established grade-level mathematics expectations (GLEs) that are linked to state assessments (Linn, Baker, & Betebenner, 2002) in an attempt to focus mathematics instruction and improve student test scores.

Post-NCLB, states are required to participate in the National Assessment of Educational Progress testing program in reading and mathematics every other year in order to provide comparisons across states. The National Assessment of Educational Progress (NAEP) is the only national representative assessment of what students know and can do in various subjects. Also referred to as the “Nation’s Report Card,” its purpose is to provide policy makers and the public with information regarding student achievement. The intent is to provide information on a representative sample across the United States. The federal government now uses NAEP results for assessing state achievement levels.

The content of the NAEP mathematics assessment is not tied to curriculum but rather to a mathematics framework that describes the important mathematical knowledge and skills that are to be assessed. This framework identifies five major content strands: (1) number sense, properties, and operations; (2) measurement; (3) geometry and spatial sense; (4) data analysis, statistics, and probability; and (5) algebra and functions. Along with these content areas, NAEP assesses students’ overall ability to use mathematics in three ability levels: (1) conceptual understanding; (2) procedural knowledge; and (3) problem solving. The target populations include students at Grades 4 and 8 (and in some years Grade 12). Student achievement is summarized in four levels: (1) below basic, (2) basic, (3) proficient, and (4) advanced.

Examining NAEP results over time provides an opportunity to examine achievement trends. From 1990-00, NAEP results showed continual improvement in student mathematics scores at Grades 4 and 8, but the achievement gaps between White students and their Black, Hispanic, and American Indian peers were wide and did not improve at either grade level. The overall mathematics scores in 2003 increased significantly over those in 2000, and the size of the achievement gaps also decreased (Lubienski & Crockett, 2007). Despite the decrease in the achievement gap, severe disparities between racial groups in achievement remain. The 2003 data indicate the percentage of students reaching basic, proficient, and advanced levels increased (Kloosterman & Lester, 2007). Furthermore, students at Grades 4 and 8 scored higher than the international average on the Trends in International Mathematics and Science Study (TIMSS) test. There is, therefore, some basis for optimism about the direction of mathematics achievement in the United States.

The mathematical performance of students is a highly visible area of interest. A series of international studies in mathematics conducted by the International Association for the Evaluation of Educational Achievement (IEA) has prompted extensive discussion of student achievement. The First International Mathematics Study (FIMS) was conducted in the 1960s and the Second International Mathematics Study (SIMS) followed in the 1980s. TIMMS, now known as the Trends in International Mathematics and Science Study, currently administers an assessment in mathematics every four years and targets two student populations: ages 9-10 and ages 13-14.

The focus of the TIMSS studies is measuring the cur-ricular knowledge that students have learned in their respective programs. Subject matter specialists from all countries participating in the study contribute to the test development. TIMSS is similar in content to the NAEP assessment and it addresses five main strands of mathematics: number, measurement, geometry, data, and algebra.

Internationally, the United States is not in the top tier of countries at any grade level, although fourth-grade students fare better in these comparisons than do eighth-grade students. In 2003, Grade 4 students performed significantly above the international average outperforming 13 of the other 24 participating countries. Grade 8 students also exceeded the international averages outperforming 24 of the 44 other participating countries (Gonzales et al, 2004). In contrast to NAEP, TIMSS scores for U.S. students did not change significantly between 1995 and 2003 for fourth grade (Kloosterman & Walcott, 2007). For eighth grade there was an increase between 1995 and 1999 but no measurable differences between 1999 and 2003 (Gonzales et al., 2004).

International studies continue to receive attention with the latest information available coming from the Programme for International Student Assessment (PISA) studies of students at the school-leaving age. The purpose of PISA is different than that of NAEP and TIMSS. PISA is designed for measuring mathematical literacy, reading literacy, and science literacy in 15-year-old students, the school-leaving age in many countries. In the last PISA assessment in mathematics (2003), general mathematics literacy and problem solving were tested separately. U.S. students performed significantly below the international average on both. For more detailed information on U.S. student performance see http://www.pisa.oecd.org.

## Conclusion

Curriculum, teaching, and assessment all play influential roles in the mathematics students learn in the classroom. Curriculum controls what content students have the opportunity to learn, and thus should act as a vehicle for focusing attention on the most important mathematical ideas to be taught. Care must be exercised to ensure that student opportunities to learn are not limited to trivial mathematics and that students are challenged to learn to the full extent of their ability. Teaching determines both the quality and depth of the mathematical knowledge students acquire. This, in turn, directly affects whether students are able to use their mathematical knowledge in productive ways in their personal lives and in the workplace. Finally, assessment done well at the classroom, state, and national levels provides data useful for decision making that can improve all aspects of students’ mathematical education.