This Critical Issue was researched and written by Lisa Palacios, senior program associate at Learning Point Associates.
Editorial guidance was provided by Gilbert Valdez, Ph.D., director of the North Central Regional Technology in Education Consortium and codirector of the North Central Eisenhower Mathematics and Science Consortium (NCEMSC); and Barbara Youngren, director, NCEMSC.
The Critical Issue team would like to acknowledge the following experts for reviewing this article: Arlene Hambrick, Ph.D., National-Louis University faculty member and a private consultant in the field of equity in mathematics and science education; Debi Maddox, coordinator at School City of Hammond, Indiana; and Larry Osthus, mathematics consultant.
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Since the move toward national and state standards began in the late 1990s, national educational standards have been developed in most subject areas. The National Council of Teachers of Mathematics (NCTM) has outlined principles and standards for the discipline of mathematics. Their Principles and Standards for School Mathematics, published in 2000, reflects a concern that students in the United States are not mathematically literate and often fail to see the relationship between mathematics learned in school and real-life situations.
The Principles and Standards for School Mathematics are expressed as six principles (equity, curriculum, teaching, learning, assessment, and technology) and 10 standards (five content and five process standards: number and operations, algebra, geometry, measurement, data analysis and probability; problem solving, reasoning and proof, communication, connections, and representation) (NCTM, 2000c). Schools and communities, with guidance from the standards document, must determine how to embed these standards and principles into their schools' mathematics instruction. Furthermore, the No Child Left Behind (NCLB) Act (2002), reauthorization of the Elementary and Secondary Education Act, presents schools with a huge challenge: To offer rigorous curriculum content, instructional methods, and forms of assessment that not only are aligned with state standards and/or the NCTM Principles and Standards for School Mathematics, but also to ensure that their students score well and continue to make adequate yearly progress (AYP) on state assessments.
This Critical Issue presents an overview of mathematics standards and assessment issues. It also offers valuable accountability information to individuals, both those very experienced and knowledgeable about mathematics standards and novices. The Critical Issue begins with background information so that the reader can appreciate what is at stake. It then addresses the question of "how do we know whether students are reaching the standards through the assessment given and the interpretation of those assessments?" It takes the reader through what parents, students, and professionals should know is important in the management and application of standards-based instruction and to accomplish the goals of NCLB. The Critical Issue also provides different points of view of the critics. The key question addressed in this issue is: What approaches are necessary to further integrate Principles and Standards for School Mathematics into classroom practice to improve mathematics instruction and meet the accountability outlined by the NCLB legislation?
A Brief History of Mathematics Standards Development
The move toward national and state standards for mathematics instruction and outcomes was given major impetus with the publication of a 1989 report on the implications of then current practice. According to Everybody Counts: A Report to the Nation on the Future of Mathematics Education (Mathematical Sciences Education Board [MSEB], 1989), American children have mathematical skills that are insufficient for problem solving in the workplace and for mathematical literacy at the college level. The report states, "Current mathematical achievement of U.S. students is nowhere near what is required to sustain our nation's leadership in a globally technological society" (MSEB, 1989, p. 1).
In response to this growing concern about students' mathematical understanding, NCTM published Curriculum and Evaluation Standards for School Mathematics (Suydam, 1990). In publishing this breakthrough document—the first set of national standards in a major content area—NCTM began a discussion about what students should know and be able to do in certain grade levels (K–4, 5–8, and 9–12). Following the publication of curriculum standards, NCTM published in 1991 the Professional Standards for Teaching Mathematics and followed with Assessment Standards for School Mathematics in 1995. These two documents expanded upon the standards-based movement by including models for both teaching and assessment.
However, because of considerable criticism of the first set of standards for not paying sufficient attention to basic skills, NCTM started a revision. The second version added language emphasizing basic skills. The revised standards were published in 2000 as Principles and Standards for School Mathematics. This new set of standards, while built upon the foundation of the previous standards document, differs from the original in important ways. Primarily, it integrates the classroom-related portions of all three prior works and reorganizes standards content into four grade bands: prekindergarten through Grade 2, Grades 3–5, Grades 6–8, and Grades 9–12. According to NCTM (2002), the new standards document "keeps the direction and core messages of the original standards documents while updating the information to reflect current research on mathematics teaching and learning, the wisdom we have gained in more than 10 years of promoting standards-based reform, and advances in technology" (p. 1). The new edition of standards includes the following (NCTM, 2002, pp. 1–2):
To support the research base of the revised mathematics standards, NCTM published A Research Companion toPrinciples and Standards for School Mathematics (Kilpatrick, Martin, & Schifter, 2003). The research companion addresses the issues of research and its role in setting the standards in the mathematics classroom. The comprehensive analysis provides the reader with current thinking about school mathematics education. The book is divided into three sections discussing research on principles and standards, perspectives on teaching and learning, and the creation of the mathematics standards. Topics covered in the research companion include assessment, mathematical power, professional development, and the content standards. The articles are written by multiple experts and noteworthy authors such as Deborah Ball, Joan Ferrini-Mundi, Robert Siegler, Patrick Thompson, and Erna Yackel.
The other source for the research base is the article, "Scaling Up Innovative Practices in Mathematics and Science," written by Thomas Carpenter and his colleagues (2004) at the National Center for Improving Student Learning and Achievement in Mathematics and Science (NCISLA), a former mathematics and science center located at the University of Wisconsin–Madison. Their eight years of classroom research shows there are three key instructional practices that are fundamental in allowing new visions of mathematics and science to become the norm in the classroom. Those practices—modeling, generalization, and justification—are supported by long-term research as critical yet still invisible practices necessary for students to be successful and productive in learning mathematics and science.
Assessing Standards-Based Mathematics: NAEP, TIMSS, and NCLB
Although most American students can demonstrate mastery of basic mathematics facts, many are not capable of using this knowledge to solve everyday problems. Standards-based reform has many curricular and instructional prerequisites, which require teachers to use specific knowledge and skills and to know what works in mathematics education.
Knowing what works in education requires teachers be knowledgeable about research-based materials. The What Works Clearinghouse was established in 2002 by the U.S. Department of Education's Institute of Education Sciences to provide educators, policymakers, researchers, and the public with a central and trusted source of scientific evidence of what works in education. The clearinghouse currently features mathematics middle school curricula research studies.
In this new era of accountability under NCLB, alignment with state standards is an important criterion with major impact on definitions of quality data. "States will be held accountable for deliverables (e.g., adequate yearly progress determinations, annual report cards, [and] diagnostic assessments aligned with academic standards and linked to the state's assessments," said Glynn Ligon (2004, p. 1), president of ESP Solutions Group, at the Empowering Accountability and Assessment Using Technology Summit 2004. The summit highlighted the need for adequate information systems as well as the need to collect and report quality data.
NAEP Mathematics Assessment
For more than 20 years, the National Center for Educational Statistics (NCES) has used the National Assessment of Educational Progress (NAEP) to monitor the educational achievement of American students and changes in that achievement across time. Through random-sampling testing, NAEP results help us to understand the extent to which our students can solve problems with their mathematical knowledge. According to NCES:
"[NAEP is the] source for information on mathematics and science achievement at key stages of education across the country using nationally established external benchmarks of performance (e.g., basic, proficient, advanced). The frameworks and benchmarks are established by the National Assessment Governing Board (NAGB) and are based on the collaborative input of a wide range of experts and participants in the United States from government, education, business, and public sectors. Ultimately, they are intended to reflect the best thinking about the knowledge, skills, and competencies needed for students to have an in-depth understanding of these subjects at different grades." (Scott, 2004, p. 2)
In the past, the NAEP test was administered to schools on a voluntary basis. Under NCLB legislation, schools accepting Title I funds must agree to administer the NAEP test if they have been randomly selected.
Following the NCTM mathematics standards publications, NAEP published an assessment framework for mathematics called the NAEP Mathematics Framework. This 1996 document, developed by the National Assessment Governing Board (NAGB), describes the skills and content measured in the assessment. The NAEP mathematics assessment was developed using this framework. The 1996 mathematics framework reflected then-current curricular emphases and objectives, yet still maintains a connection with the 1990 and 1992 assessments (NCES, 2004). It focuses on five broad strands of mathematics content: (1) number sense, properties and operations; (2) measurement; (3) geometry and spatial sense; (4) data analysis, statistics, and probability; and (5) algebra and functions (NCES, 2004). The 1996 framework also includes three types of mathematical abilities: conceptual understanding, procedural knowledge, and problem solving.
Since the Curriculum and Evaluation Standards for School Mathematics was published in 1990, NAEP mathematics assessments have continually been updated as new mathematics standards evolved and have focused on fostering mathematical knowledge and power in the nation's students (NCES, 2004). Mathematical power is defined as consisting of a few elements. According to NAGB:
"Mathematical power is characterized as a student's overall ability to gather and use mathematical knowledge through exploring, conjecturing, and reasoning logically; solving nonroutine problems; communicating about and through mathematics; and connecting mathematical ideas in one context with mathematical ideas in another context or with ideas from another discipline in the same or related contexts." (Orrill & French, 2002, p. 35)
The 1996, 2000, and 2003 nationwide mathematics assessments focused on reasoning and communication by requiring students to connect their learning across mathematical content strands (NCES, 2004). Figure 1, taken from the Mathematics Framework for the 2003 National Assessment of Educational Progress publication, shows the relationship among the mathematical abilities, content strands, and power as described in the mathematics framework (NCES, 2004; Orrill & French, 2002, p. 11).
Figure 1. Framework for the 1996, 2000, and 2003 Mathematics Assessments
Based on NCES (2004), the mathematics framework specifies the percentage distribution of items to be devoted to each content strand by Grades 4 and 8. T he level of mathematics proficiency is determined for students in Grades 4, 8, and 12. NAGB's Loomis & Bourque (2001) provide a document, titled National Assessment of Educational Progress Achievement Levels 1992–1998 for Mathematics, with descriptions of the NAEP mathematics achievement levels (basic, proficient, and advanced), cut scores, and sample questions with illustrations of student knowledge and skills required within each achievement level. According to the document, which could well serve as a reference booklet for parents and teachers interested in learning about the NAEP assessment, the following definitions of the achievement levels in Table 1 apply to all subjects and all grades assessed by NAEP.
Table 1. NAEP Achievement-Level Policy Definitions
Basic denotes partial mastery of prerequisite knowledge and skills that are fundamental for proficient work at each grade.
Proficient represents solid academic performance for each grade assessed. Students reaching this level have demonstrated competency over challenging subject matter, including subject-matter knowledge, application of such knowledge to real-world situations, and analytical skills appropriate to the subject matter.
Advanced represents superior performance.
The 2003 NAEP assessment has brought encouraging news about student achievement in at least a few levels. Regarding mathematics results, Stevens (2003), an NAGB member, claimed, "Something positive is happening in our schools" (p. 1). Indeed, results of the 2003 NAEP (NCES, 2005) show that overall mathematics scores are improving, especially in the fourth grade. It is significant that this improvement coincides with the growth of the standards movement and the fact that elementary students are exposed to all NCTM standards, where before the emphasis in mathematics instruction was primarily on computation.
As Figure 2 shows, growth is seen in both Grades 4 and 8.
Figure 2. Average Mathematics Scale Scores, Grades 4 and 8: 1990–2003
Source: NCES, 2004: http://nces.ed.gov/nationsreportcard/mathematics/results2003/natscalescore.asp
The graph shows the upward trend in scale scores over a 13-year period for Grades 4 and 8, both for students allowed accommodations (changes in testing conditions as required by individualized education programs) and not. In fourth- and eighth-grade mathematics, the scores in 2003 were higher than in all previous assessment years. The increases are evident at all three levels of student proficiency, as seen on the NAEP Mathematics Achievement Levels Web page (NCES, 2002).
Stevens (2003) called the achievement results at the Basic level "real progress" (p. 3). Yet he also raises the question of how the nation's students could reach the next level—proficiency. "To reach Proficient on NAEP, students must be able to apply the math they've learned to different, often unfamiliar situations, to set up problems as well as to solve what they are given," says Stevens (2003, p. 3). Even though the results do show some increases at the Proficient level in both fourth and eighth grades, the concern is with the racial gap that exists at the achievement level: "The proportion of black fourth graders reaching Proficient has climbed from just 1 percent in 1990 to 5 percent in 2000 to 10 percent in 2003. In the last three years the proportion of Hispanics reaching this level rose from 7 [percent] to 16 percent. But among white fourth graders, 43 percent now reach the Proficient level; among Asians it's 48 percent" (Stevens, 2003, p. 4).
Ensuring equity and excellence lies at the core of systemic reform efforts. The core of educational equity is to ensure that every student has access to challenging curriculum that supports his or her personal, academic, and professional goals. Regardless of differences of race, ethnic group, gender, socioeconomic status, geographic location, age, language, disability, or prior academic achievement, all students deserve equitable access to challenging and meaningful academic learning and achievement. However, access is often not enough. Equally important factors are high expectations and strong support (NCTM, 2000c). Such integrated concepts of equity have been progressively developed, defined, and articulated to school communities across the country. NCTM highlights the following components of the Equity Principle:
NCREL's Pathways to School Improvement Critical Issue "Remembering the Child: On Equity and Inclusion in Mathematics and Science Classrooms" (Hambrick & Svedkauskaite, 2004) provides a definition of equity and an overview of the issues in mathematics and science education. The overview is supported and enhanced with links to many additional resources and includes suggestions for teacher professional development, goals, action items, and managing implementation pitfalls. The article discusses equity issues regarding student groups singled out in NCLB: students who are socioeconomically disadvantaged are from major racial and ethnic groups, have disabilities, or have limited English proficiency. Additional information on how states and regions are closing the achievement gap can be found on NCREL's Closing the Achievement Gaps Web site (n.d.). Sample mathematical problems are available on the NAEP Web site (NCES, 2004).
Similar to NAEP, the Trends in International Mathematics and Science Study (TIMSS) also measures mathematics proficiency at the fourth and eighth grades, using sample-based assessments. While NAEP illustrates American students' progress in mathematics, TIMSS illustrates America's progress compared with other nations.
TIMSS provides international comparative information on mathematics achievement in the primary and middle grades in the United States:
"Like NAEP, its assessments are based on collaboratively developed frameworks for topics from curricula in mathematics … to be assessed; but unlike NAEP, the framework and related consensus process involves content experts, education professionals, and measurement specialists from many different countries." (Scott, 2004, p. 2)
In 2003, comparisons of the mathematics and science achievement of fourth graders were made among 25 participating countries. U.S. fourth graders exceeded the international averages in both mathematics and science. In mathematics specifically, U.S. fourth graders outperformed their peers in 13 of the participating countries (NCES, 2003). Students in three countries—Chinese Taipei, Japan, and Singapore—outperformed U.S. fourth graders in both mathematics and science (NCES, 2003).
Among eighth graders, however, the comparisons were made among 45 participating countries. Similar to the fourth graders, U.S. students exceeded the international average in mathematics and science. They outperformed their peers in 25 countries in mathematics and 32 countries in science (NCES, 2003). Students in seven countries—Chinese Taipei, Estonia, Hong Kong SAR, Hungary, Japan, Korea, and Singapore—outperformed U.S. eighth graders in mathematics and science (NCES, 2003).
According to Plisko (2004), associate commissioner of NCES, "TIMSS provides an important external perspective on the performance of U.S. students" (p. 5) TIMSS as well as NAEP assessments have the most in common of all in terms of content and cognitive demand (Plisko, 2004).
Assessment in the Era of NCLB
Although NAEP and TIMSS measure student achievement, NCLB requires each state to demonstrate accountability through making AYP. Because of this new requirement, all states are working to design and implement yearly tests in reading and mathematics from Grades 3–8, and a test to be administered once between 10th and 12th grades. In addition, each state must administer assessments in reading and math at three grade spans (3–5, 6–9, and 10–12). Science tests must be developed and administered during the three grade spans by 2007–08. Testing performed by NAEP and testing required by NCLB do not coordinate well in terms of test years and yearly testing of all grades. NAEP administers tests in Grades 4, 8 and 12. TIMSS operates on a similar schedule. For the latest information on NAEP and TIMSS and their relevance to NCLB, visit the NAEP and No Child Left Behind section of the NAEP Web site.
NCLB also has set similar requirements to those of NAEP for standards and proficiency, yet the legislation leaves up to the states how to articulate each level of achievement. According to NCLB (2002), academic content standards should be challenging and "specify what children are expected to know and be able to do, contain coherent and rigorous content, and encourage the teaching of advanced skills" (Part A, Sec. 1111(b)(D)(i)(I–III). They should align with each state's academic content standards; "describe two levels of high achievement (proficient and advanced) that determine how well children are mastering the material in the state academic content standards; and describe a third level of achievement (basic) to provide complete information about the progress of the lower-achieving children toward mastering the proficient and advanced levels of achievement" (NCLB, 2002, Part A, Sec. 1111[b][D][ii][I–II]).
One way that the United States has addressed American students' progress in mathematics as well as other subject areas has been to develop a nationwide system of academic standards that define content and assessments. NCLB has formalized this system and strengthened Title I accountability by requiring states to implement statewide accountability systems covering all public schools and students.
Challenging Academic Content Standards
The NLCB-mandated accountability systems must be based on challenging state standards in reading and mathematics, annual testing as described previously, and annual statewide progress objectives ensuring that all groups of students reach proficiency within 12 years (i.e., by 2014). To ensure that no group is left behind, assessment results and state progress objectives must be broken out by poverty, race, ethnicity, disability, and limited English proficiency.
Forty-nine of the 50 states have adopted state-level standards in mathematics. Ohio provides an excellent example of how national standards and state standards can be correlated (Ohio Department of Education, n.d.). The Ohio Department of Education Web site provides academic content standards in all subject areas, an alignment toolkit, and a district curriculum alignment tool so that individual districts can create their own standards documents based on the state standard subsets. Additionally, Ohio has compiled a model curriculum made up of standards-based mathematics lessons in all standards and all grade levels. Ohio also provides its constituent districts statewide with research-based opportunities for mathematics and science professional development (i.e., Ohio Mathematics Academy Program and Ohio Science Institute) developed by Miami University, the Ohio Department of Education, and the Science and Mathematics Network.
Assessment is a central focus of the standards movement and is perplexing to many teachers. Often teachers feel formal assessment and testing take too much time from their instruction. But they feel pressured to learn to incorporate as much assessment analysis as they can into their instructional methods. Combined with pressure to teach to the standards, teachers may often feel they are teaching to the test.
To alleviate the tension teachers feel about assessment, it is important for them to know the specific short- and long-term goals and purposes of assessment. There are several purposes of assessment in mathematics education. They include monitoring student progress, planning instructional activities, and formulating education policy (Champagne, Sherwood, & Cezikturk, n.d.). The Assessment Standards for School Mathematics (NCTM, 1995) presents six standards for mathematics assessment In Principles and Standards for School Mathematics, NCTM concluded that exemplary mathematics assessment based on those standards should do as follows (NCTM, 2000c, p. 22):
Adequate Yearly Progress (AYP)
NCLB requires that each state defines AYP for school districts and schools within the parameters set by NCLB (U.S. Department of Education, 2004). The process for defining AYP is as follows:
"Each state sets the minimum levels of improvement—measurable in terms of student performance—that school districts and schools must achieve within time frames specified in the law. In general, it works like this: Each state begins by setting a starting point that is based on the performance of its lowest-achieving demographic group or of the lowest-achieving schools in the state, whichever is higher. The state then sets the level of student achievement that a school must attain [after two years] in order to [continue to show] AYP. Subsequent thresholds must increase at least once every three years, until, at the end of 12 years, all students in the state are achieving at the proficient level on state assessments in reading and language arts [and] math." (p. 11)
School districts and schools that fail to make AYP toward statewide proficiency goals will, over time, be subject to improvement through corrective action and restructuring measures aimed at getting them back on course to meet state standards. Schools that meet or exceed AYP objectives or close achievement gaps will be eligible for state academic achievement awards: States may financially reward teachers in schools that receive academic achievement awards (U.S. Department of Education, 2004).
Administration and teachers share the responsibility for AYP. The failure of one subgroup to make AYP can cause the school to be labeled as needing improvement. This label means that all students in the school are offered the opportunity to transfer to a different school in the district, with transportation provided. If improvement is not seen (i.e., AYP is missed again), low-achieving students become eligible for supplementary educational services (such as tutoring) at district expense.
Finally, the school may be required to replace certain staff, fully implement a new curriculum, or, in the end restructure the school either as a charter school, replace all staff, or turn the school over to the state or a private company. (View NCLB regulations on AYP.)
Such stringent AYP processes require school and district administrations to find the most effective interventions for low-achieving students in order to best use, and in some cases retain, their federal Title I dollars, in addition to ensuring that their current staff and new hires meet the definition of highly qualified.
Highly Qualified Teachers
NCLB requires highly qualified teachers in each classroom. To be deemed highly qualified, teachers must have the following:
Teachers (in middle and high school) must prove that they know the subject they teach with one of the following:
NCLB allows states to develop an additional way for current teachers to demonstrate subject-matter competency and meet highly qualified teacher requirements. Proof may consist of a combination of teaching experience, professional development, and knowledge in the subject garnered over time in the profession. However, time teaching a given subject alone cannot be used to qualify as "highly qualified."
These requirements can be daunting for schools and districts. Teachers at the middle and high school levels who find that they are not considered highly qualified as described above may find themselves unable to obtain a teaching position, released before achieving tenure, or if tenured, moved to teach a subject in which they are deemed highly qualified even if they have taught mathematics for years.
Teachers in Community Unit School District 200 in Wheaton, Illinois, use a number of ways to stay current in the teaching field.
Gabriela Castillo, a first-grade bilingual teacher at the district's Johnson Elementary School, says ways such as methods or mathematics courses offered through the district, collaboration with fellow teachers, and teacher observations can help her sharpen her instructional skills and professional knowledge. [Video: 1:00]
Teachers and school districts have already found it a challenge to design curricula, instruction, and assessments that meet, exceed or coordinate national, state, and local district standards. The challenge does not end there because NCLB accountability requirements mean that schools and districts must quickly implement best practice as determined by scientifically based research in mathematics instruction while ensuring that all teachers meet the definition of high quality.
Meeting Accountability Requirements
With the advent of NCLB, all of these accountability factors form the basis for educational success, while high-stakes tests determine students' and schools' educational future. As NCTM (2000b) states, such tests "are used to make significant educational decisions about children, teachers, schools, or school districts" and make "the determination of such things as graduation, course credit, grade placement, promotion to the next grade, or placement in special groups" (p. 1). Such an accountability system makes it imperative to implement NCTM principles and standards in curriculum, instruction, and assessment as well as to better understand the needs of the NCLB identified subgroups.
To summarize, NCLB requires the following:
To meet the NCLB requirements in regards to the subject matter of mathematics, NCTM (2002) recommends the following key elements.
Students Should Learn and Experience Mathematics in a Standards-Based Classroom. Standards-based instruction allows teachers and students to be on the same page by specifying how teachers and students will meet their education goals, including specific concepts, order, or instructional materials (Krueger & Sutton, 2001). In standards-based instruction, standards delineate what matters, provide clarity and a fixed point of reference for students and teachers, guide instruction so that it is focused on student learning, provide a common language to have conversations, help ensure equal educational opportunities, assist in identifying struggling students, and meet federal guidelines (Ohio Department of Education, n.d.).
Because national mathematics standards are voluntary and do not prescribe a single approach to teaching the subject, it is up to the individual local educational entities to determine the mathematics course content, organization, focus, and delivery. In the classroom, students should be learning how to apply their mathematics knowledge to the real world and share their learning processes with fellow students (NCTM, 2002). As they learn mathematics, students should use a range of mathematical and technological tools such as manipulatives, calculators, and computers (NCTM, 2002). The role of the teacher is to probe student thinking by posing problems, asking questions, and encouraging exploration of solutions (NCTM, 2002). One of the challenges with standards is that even though they do provide a direction for learning, they may sometimes be too rigorous or become the sole measure of how students learn. There may be danger in allowing standards to entirely drive the curriculum.
Students Need Multiple Opportunities to Apply Basic Skills to Problem Solving. Students inevitably have to solve problems in their daily lives. In the mathematics classroom, the essential task is to have students problem solve in ways that apply and challenge their basic skills. This process of helping students connect basic skills to the problem-solving process should start as early as the first grade.
The emphasis should be on building students' mathematical power (i.e., enhancing their ability to reason, problem solve, make connections, communicate with their fellow students, and use representations in their solutions) (NCTM, 2002). In the classroom, problem solving comes first and learning basic skills becomes a part of what it means to problem solve. Fostering both computational skills and conceptual understanding will allow students to solve problems that they encounter in their daily lives (NCLB, 2002). This skill is critical for all students, including those with limited English proficiency.
Students Should Be Empowered to Use a FullRange of Technological Tools. Technology has considerable potential for increasing interest in, and improving the quality of learning in, the mathematics classroom. Effective use of technology is possible only if sufficient attention is given to creating and coordinating appropriate curricula, instructional pedagogy, and assessments. Such effort needs to be supported by classroom access to sufficient technology, access to the Internet, and, especially, the ability of the teacher to model the necessary uses of the technology.
Gilbert Valdez, Ph.D., codirector of the North
Central Eisenhower Mathematics and Science Corsortium, says technology
adds productivity to learning. [Video: 1:20]
Students Should Be Provided with Assessments that are Ongoing, Continual, and Multifaceted. An earlier tendency in the United States has been to implement standardized testing as a means to improve schools, starting with a district level and going up to an international level (Stiggins, 2004). The mistake, it appears, was to "believe that once-a-year standardized assessments alone can provide sufficient information and motivation to increase student learning" (Stiggins, 2004, pp. 22–23). A single annual test is grossly inefficient. Teachers must also be equipped and encouraged to use formal and informal formative assessment year-round to drive their teaching in ways that influence student habits inside and outside the classroom, and ideally, in the long run. As Dr.
Valdez says states need to recognize that learning is about big concepts and something that has meaning for a lifetime . [Video: :36]
While summative assessment is used for accountability to showcase and prove that students gained knowledge over a certain time, formative assessment is valuable in its intent to improve learning and to change instruction based on results. Therefore, it is important to know the different purposes and intents of both assessments.
According to Dr. Valdez, the two tests have two different, purposes and, and intents. [Video: :30]
NCTM (2002) advocates that students be given ample opportunities to demonstrate their mathematical understanding through a variety of assessment techniques such as portfolios, discussions, presentations, and projects, in addition to the traditional approach of written tests. NCREL's Pathways to School Improvement Critical Issue "Multiple Dimensions of Assessment that Support Student Progress in Science and Mathematics" (NOTE: please hyperlink to the article when it is posted) provides more information on assessment issues from various perspectives.
What Do These Recommendations Mean for Classroom Teachers?
The creation and delivery of such "students should" and "students need" recommendations cannot happen without dedicated teachers. However, these NCTM instructional recommendations are dependent on teacher willingness, preparation, and support. To create and sustain these classroom learning experiences, teachers need to do the following:
Classroom Strategies for Meeting Standards
A mathematics classroom that is standards based should have four embedded central strategies, according to an ENC Focus issue (ENC, 2004) "Looking Into a Standards-Based Classroom":
However, in an elementary classroom, teachers are often challenged by the need for their own comfort in using technology; enhancing students' problem-solving skills and critical thinking with their knowledge of basic skills; employing constant assessment; and, last but certainly not least, their own need for broad and deep knowledge of mathematical content. Indeed, the research is showing that the more elementary school teachers understand the mathematical content behind what they teach, the greater student achievement is and the less likely students are to develop misconceptions that interfere with their learning in later years. Teachers who know the mathematics content more deeply are more likely to skillfully guide class discussions. Teachers' sound conceptual base is the hallmark of standards-based classroom learning because it supports the process standards of Principles and Standards for School Mathematics. Taking time to enhance the mathematics-content knowledge of elementary teachers is problematic because they are also responsible for teaching all the other content areas as well. However, the evidence suggests that this would be time well spent.
Based on the message from the Principles and Standards for School Mathematics that all the process skills be taught at every grade level, all teachers should do the following:
This is particularly challenging in classrooms that operate on a bell schedule, which segments the day into less than one-hour class periods. Teachers in that situation may only teach mathematics; however, they have many more students to teach.
Overall, all teachers of mathematics, whether at the elementary or secondary level, have to be aware of the national, state, and local mathematics standards and principles as well as best practice in teaching and assessment. Teaching in a standards-based classroom could require huge changes in a teacher's approach to the students.
In the era of NCLB, mathematics education should encompass the following goals:
These goals present an enormous challenge: to create and implement a comprehensive and long-term plan for the improvement of mathematics curricula, instruction, and assessment in the education system. They have been thoughtfully articulated to maximize the goal of mathematical literacy for all students. However, the education stakeholders at the school, district, and state level must commit to far-reaching analysis, training, and persistent application of such actions.
What approaches or action options could these stakeholders take to ensure that NCTM's Principles and Standards for School Mathematics are integrated into classroom practice to improve mathematics instruction and meet the accountability outlined by the NCLB legislation?
The following action options are sorted by stakeholders most directly responsible for their implementation: students, teachers, administrators, and parents and community leaders. Contemplated by stakeholder, they capture the depth of the individual commitment necessary for ongoing success. Viewed as a whole, they capture the scale of the institutional effort needed to ensure mathematical literacy for all students.
To become mathematically literate, students must be empowered to do as follows:
To facilitate students becoming mathematically literate most effectively, teachers should do the following:
To be effective student and teacher facilitators, administrators should do the following:
To facilitate and enhance the learning of students, parents and community leaders must do the following:
There have been many positive changes and much progress in institutionalizing the learning guidelines and expectations of the standards movement. But with that progress has come many challenges and pitfalls for schools trying to implement standards-based instruction. In most cases, standards are a starting point for curriculum development, but by themselves are not nearly ready for direct use in classrooms because they are often very general, and yet can still take years to achieve. Further, the standards were created by advocates of mathematics education and literacy. Thus, when all are considered together, they present educators with the task of crowding into 12½ years what some experts believe is more than 19 years of learning expectation. As a result, standards are often addressed in a hit-and-miss fashion. Several other pitfalls are discussed below.
Misaligned Curriculum and Assessment
Testing and assessment programs that are not coordinated with the instructional aims and curricular goals such as those derived from Principles and Standards for School Mathematics, can be significant obstacles to implementing reform in U.S. instruction, and assessment. (See NCREL's Pathways Testing and Assessment Programs That Are Not Coordinated With the Instructional Aims and Curricular Goals.) Teachers and school district administrators often feel significant pressure for their students to perform well on these tests (especially in the era of NCLB). Yet the tests often monitor the achievement of computational skills rather than the reasoning and problem-solving abilities of students. To avoid such a pitfall, tests need to align, of course, with the curriculum. However, students also should be tested for higher skills that align with district, school, and classroom goals.
There are major efforts under way to incorporate such assessment practices. Teachers and administrators can lead in advocating for such changes by seeking positions on state panels that review the development of state assessments and review items. Recent research summarized by the National Research Council in two publications—Adding It Up: Helping Children Learn Mathematics and How Students Learn: History, Mathematics and Science in the Classroom—can help identify best practices in learning strategies for students, and mathematics concepts and skills that can be taught more effectively to provide stronger foundations for later mathematical learning.
Mathematics Taught in Isolation From the Real World
Another pitfall is the pervasive practice of teaching the procedures of mathematics detached from the meaning and applications of these procedures in the real world. There are a host of reasons for linking procedures-instruction consistently and effectively to real-world application. (Many are discussed elsewhere in this report.) Schools and classrooms moving toward more of an emphasis on teaching students both what to do and why will want to explore the roadblocks to this practice. Marilyn Burns, a founder of Math Solutions Professional Development, shares her ideas about helping children learn in the mathematics classroom; talks about how teachers can learn to really understand mathematics; and suggests ways to help parents understand the role of the NCTM standards in the classroom (Herrera, 2005).
Short-Term and Unsustained Professional Development
Many curriculum materials are implemented with only one or two days training from the publisher. Then principals and district administrators, too often, expect that the curriculum will be successfully implemented. Both the research into implementation of standards-based curriculum materials and the standards of the National Staff Development Council indicate that ongoing, sustained, and coherent professional development is needed to successfully implement a curriculum, align it to state and district standards, and generate fidelity to the curriculum. A curriculum series cannot be blamed for lack of student achievement if there is no indicator that series has been faithfully implemented, taught as intended, and that assessments used for student achievement align with both the curriculum materials and the intended learning outcomes as indicated by state and district standards.
Insufficient Attention to Student Subgroups
Not reaching all students could be a significant pitfall, especially students who fall into subgroups such as English language learners, students with disabilities, students from racial/ethnic backgrounds, and those who are economically disadvantaged. Teachers and administrators who are unaware of best practice teaching and learning strategies for specific subgroups are less likely to attempt to reach all students' learning needs, and either alter their teaching practices, or inquire about additional help.
Underutilized and Undertrained Use of Technology
Instructional integration of technology is more difficult than it may seem. For smooth integration, a district's technology must be available, maintained, and deployed consistently across classrooms and district in ways that make instructional sense. Further, integration efforts require software readily available that was thoughtfully chosen to integrate with curriculum. Equally important is that teachers need to have significant amounts of professional development on the use of the available technology. Teachers are often running into challenges, including a lack of student access to technology at home, schedule conflicts in computer labs, limited interaction with colleagues, lack of focus on technology within curriculum materials, and simply, overall lack of technology knowledge. Using calculators with students is a common, yet underused way to integrate technology into mathematics assessment. However, calculators should be used appropriately, effectively, and in a developmental way (e.g., for functions at the K–5 level, for fractions at the 4–6 grade level, and graphing calculators at the middle and high school level).
DIFFERENT POINTS OF VIEW:
In the era of NCLB, it is easy and comfortable to rely solely on curriculum and instruction that are determined by the high-stakes assessment tests. When parents, administrators, and teachers focus only on meeting AYP, they find it easy to advocate abandoning a standards-based system that includes content and processes that have not been tested.
Some educators dismiss the idea of standards and feel it runs counter to the goal of enhanced professionalism, since it takes the decision-making power over what is taught out of the hands of teachers and into the hands of a national standards-setting body. They feel standards, at least when used in a rigid and highly prescriptive manner, are anathema to professional autonomy. Thus, they conclude that the attempt to set forth standards in mathematics curriculum, instruction, and assessment—nationally centralized or not—is misguided.
Further, questions have been raised about whether the NCTM Principles and Standards for School Mathematics extol the right standards. Some dissenters claim that the Principles and Standards for School Mathematics has an inadequate research base, that fluency with algorithms and computational procedures is not adequately emphasized, and that the expansion in content is unnecessary. This position was carried out in public view in California in the late 1990s. California went so far as to develop its own framework and standards, claiming that basic skills were being ignored. A strong "overemphasis on pedagogy at the expense of mathematical content knowledge spawned several new curricula" (Wu, 2001, p. 1). Washington has a Web site titled The Educational Deform Watchdog Page, which states opinions on mathematics instruction, including technology use, assessment, and even block scheduling.
Critics also state there is too much emphasis on technology and calculator use, too much estimation to the detriment of calculation, not enough early emphasis on fractions, and too much emphasis on data collection. This issue merited a response from NCTM to a Wall Street Journal article in 2000, which stated that "The majority of experts on elementary-school learning maintain that, for students who lack basic number proficiency, calculators may provide only the illusion of progress" (p. 1). NCTM's then-President Lee Stiff, responded to it as a "troubling, and rather one-sided picture of the use of calculators in math education" (NCTM, 2000a, p. 1).
Eisenhower National Clearinghouse for Mathematics and Science Education
The Ohio State University
1929 Kenny Road
Columbus, OH 43210-1079
National Academy of Sciences
500 Fifth St. N.W., 11th Floor
Washington, DC 20001
National Center for
Institute of Education Sciences
U.S. Department of Education
1990 K St. N.W.
Washington, DC 20006
National Council of Teachers of Mathematics
1906 Association Drive
Reston, VA 22091-1593
Trends in International Math and Science Study
National Center for Education Statistics
1990 K St. N.W., Room 9010
Washington, DC 20006
Date posted: 2005