Implementation of the NCTM Standards

In the fall of 1992, a study group appointed by the Metropolitan Educational Research Consortium undertook a study to determine the extent to which local schools were implementing the NCTM Standards. The study group was composed of Helen Edens, Beverly H. Cook, Rosa Tapscott, James Bagby, Steven H. Lapinski, Linda H. Weber, Jacquelyn S. Joyner, Linda E. Hyslop, Bill Hoyt, Kathleen M. Cauley and John VandeWalle. This research brief summarizes the findings of three data sources: a survey of mathematics teachers, a survey of principals, and focus group interviews with selected teachers.

Purpose. The purpose of this study was to identify the progress and obstacles encountered by schools and teachers who have attempted implementation of the National Council for Teachers of Mathematics' (NCTM) Curriculum and Evaluation Standards for School Mathematics and the Professional Standards for Teaching Mathematics. Specifically, the study addressed the following questions:

1. What is the level of awareness of teachers about the NCTM Standards?

2. What is the level of implementation of the NCTM Standards in Consortium classrooms? How do the classroom practices of teachers who perceive themselves as implementing the Standards differ from other teachers?

3. What aids have helped teachers makes these changes?

4. What components are seen as hindering progress toward these changes?

The Methodology. The survey team collected data from three sources: objective responses by elementary, middle, and high school teachers (the Teacher Survey), objective responses from elementary, middle, and high school principals (the Principal Survey), and focus group discussions with a small number of teachers who reported awareness of the Standards. The Teacher Survey was piloted with teachers in the Powhatan system, and was revised based on these teachersÕ responses and comments prior to distribution to teachers in the other six MERC school systems. The survey included items to determine teacher awareness of the Standards, the frequency of use of various classroom practices, and their perceptions of aids and obstacles to implementation. Responses were received from a total of 1,892 teachers, with 55% of those surveyed responding.

THE STANDARDS

In 1989, NCTM published its now much heralded Curriculum and Evaluation Standards for School Mathematics. This comprehensive document has received virtual unanimous acceptance as a guide for curriculum and evaluation reform movements throughout the country. The Standards provide benchmark statements about specific aspects of the curriculum and about evaluation against which school districts can judge their own specific curricula. The Standards have been the major focus of mathematics education for the past four years.

Four standards, problem-solving, communication, reasoning, and connections, represent over-arching themes for the mathematics curriculum. They can be applied to nearly every area and every lesson. To teach with these four standards clearly in mind is to teach in a Standardsoriented manner.

Mathematics As Problem-Solving

According to the Standards, "problem-solving should be the central focus of the mathematics curriculum." This means much more than learning to solve word problems. Rather, mathematics as problem-solving means that problem-solving is a part of all real mathematical activity. The Standards speaks of learning a variety of general problem-solving strategies such as making a guess-andcheck or looking for a pattern. It talks about being able to formulate problems, assess results and exhibit confidence in solving problems. Problem-solving is a way of thinking and reasoning that is used in the learning and the doing of all mathematics.

Mathematics as Communication

The communication standards at each level point to the importance of being able to talk about, describe, and explain mathematical ideas. Symbolism in mathematics, along with things such as charts and graphs, should become ways of expressing mathematical ideas to others. This means that students should learn not only to interpret the language of mathematics but to use that language themselves. Learning to communicate in mathematics makes accessible the world of mathematics beyond the classroom. It also fosters interaction and exploration of ideas within the classroom as students learn in an active, verbal environment.

Mathematics As Reasoning

To reason logically is as integral to mathematics as problem-solving. In the past, logical reasoning was relegated to the tenth-grade geometry class. The Standards tells us that reasoning should be a part of mathematical activity from kindergarten on. To observe and extend a pattern, to defend a result, or to decide if an answer is correct, are all activities that involve logical reasoning. When reasoning is part of all mathematics, students learn that mathematics is not a collection of arbitrary rules but a system that makes sense and can be figured out.

Mathematical Connections

The theme of connections is really three-fold. First, the standard refers to connections within and among mathematical ideas. Addition and subtraction are intimately related. Fractional parts of a whole are connected to concepts of decimals and percents. Second, the symbols and procedures of mathematics should be clearly connected to the conceptual knowledge that the symbolism represents. Rules such as "invert the divisor and multiply" should never be learned in the absence of well developed supporting concepts. Third, mathematics should frequently be integrated with other discipline areas, with the exploration of real applications of mathematics in the real world. Children should see that mathematics plays a significant role in art, science, and social studies. Mathematics should be viewed as a meaningful and relevant discipline, in terms of both how it is done and how it is used.

OVERVIEW OF FINDINGS

1. The majority of teachers and principals are aware of the NCTM Curriculum and Evaluation Standards and are in agreement with them.

2. Approximately 21% of elementary and 53% of middle and secondary teachers report that they are implementing the Standards. The degree of implementation, however, is relatively low. This low degree of implementation is reasonable given that the Standards are a relatively recent development and that implementation requires a reconceptualization of mathematics teaching and assessment. Evidence of the low level of implementation is indicated by the following points.
   • Teachers at all grade levels rely too heavily on nonStandards oriented textbooks and the problems in them. Instead, the Standards recommend that more emphasis be given to student generated problems, Òreal-lifeÓ problems, and non-routine problems.

   • The majority of teachers do not use teaching strategies recommended in the Standards on a weekly basis. Evidence suggests that a number of strategies, such as cooperative group work or student justification of answers to problems, are being used 2-3 times a month.

   • The curriculum at the elementary and middle school levels, in contrast to the Standards, continues to emphasize computation. Areas such as statistics and probability should be emphasized more.

   • Technology, particularly computers and calculators, is underutilized at all grade levels. Teachers in the focus groups report frustration at the unavailability of appropriate calculators, computers and software. Manipulatives are not being used and/or are not readily available, especially at the middle and secondary levels.

   • Teachers who report implementing the Standards show a higher frequency of use of the recommended teaching strategies than the unchanged teachers.

   • Elementary and middle school teachers who are making changes in their teaching seem to implement the connections and reasoning themes of the Standards more readily than problem-solving or communications themes.

   • Across grade levels, writing about mathematical ideas is a weakness, both as a teaching strategy and as an assessment strategy.

   • Virtually no attempt has been made to implement recommendations for student assessment. Both the survey and the focus groups indicate that the majority of teachers do not appear to use alternative assessment techniques. The majority of teachers do not appear to distinguish between assessment for diagnosis of student understanding and assessment for grading

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only 20 to 30 percent (depending on grade level) of teachers in the changed group report that their students never work on project problems compared with 37 to 49 percent of the unchanged group. Thus, although changed teachers do not report use of project problems on a truly frequent basis, at all grade levels they are much more likely to use them, as compared with unchanged teachers.

In a similar manner, with respect to the theme of mathematics as communication, although fewer than 25% of the "changed" teachers report asking students to write about mathematical ideas two or more times a month, dramatic contrasts are also evident in the number of teachers in each group reporting that they never use this strategy. At all three grade levels, students in changed classrooms are much more likely to be asked to do at least some writing about mathematics than students in the unchanged classrooms.

In summary, the evidence suggests that motivated teachers are making more frequent use of the teaching strategies recommended in each of the four areas highlighted by the Standards, as compared with the remaining teachers, who reported no efforts to change. These differences in implementation are evident with respect to the recommendations concerning reasoning and connections (which teachers as a whole already appear to be following to a significant degree), as well as those concerning problemsolving and communication (areas in which teachers as a whole report considerable room for improvement). The magnitude of these differences is sometimes substantial even in areas in which the recommendations of the Standards diverge from traditional practices, such as journal writing and work on open-ended problems. This analysis also highlights some areas in which progress has been slow even for motivated and aware mathematics teachers, suggesting the need for additional administrative support.

AIDS TO IMPLEMENTATION

Teachers report that more staff development is needed to assist them in making the transition to a more Standardslooking curriculum. Even focus group teachers, who report that inservices are available often, do not see much value in the type that they are receiving. They are requesting more content specific staff development. They overwhelmingly approve of time to see model teachers in action and having quality time to talk and discuss about teaching ideas. Teachers see support from the principal and the guidance of lead teachers as very important. They want a principal who is well informed, who is able to discuss the Standards and who will work with mathematics teachers to find out what is required to implement them.

Comparison of changed and unchanged teachers in this area reinforces the impression that administrative support, and an active interest on the part of principals in teachers' efforts at implementation, are important sources of motivation for teachers who are working to change their instructional practices. Among elementary teachers, for example, teachers who were making changes in response to the Standards were much more likely to report that their schools or school systems had:

• designated "lead" teachers
• provided special training for these lead teachers
• revised criteria for textbook selection
• offered one or more in-services on the Standards
• maintained a library of Standards-relevant materials.

The results of this comparison for middle and high school teachers were similar to those reported for elementary teachers. Teachers who see themselves as changing in response to the recommendations of the Standards, reported substantially higher levels of active administrative support relative to teachers who do not see themselves as making such changes. This suggests that administrative changes may have a direct and beneficial effect on classroom practices in mathematics.

OBSTACLES TO IMPLEMENTATION

Time was a primary obstacle to implementation at all three grade levels. Elementary teachers described pressure to make changes in other curriculum areas, such as history and language arts, as well as in mathematics. Secondary teachers were more likely to cite lack of preparation time necessary to develop alternatives to the traditional, sequential mathematics curriculum. They also require time to coordinate with other teachers who worked with students during the prior year, or who would teach these same students the following year, to assure a coordinated sequence of instruction.

A second major obstacle, identified by teachers at all three grade levels, is the pressure to have students succeed on standardized tests. Teachers fear repercussions from administrators and parents in the event that changes in their class structure or content result in decreases in students' scores relative to national norms.

A third major obstacle is lack of resources, particularly technological aids (computers, calculators, and manipulatives). Teachers complained of having outdated equipment or severely limited access to the equipment that is available. Upper elementary and middle grade teachers do not have an accumulated supply of manipulatives and feel that these must be supplied.

GENERAL CONCLUSION AND SUMMARY

The implementation of the NCTM Standards appears to have the support of the majority of metropolitan teachers. Most are aware of the NCTM Standards, have access to a copy of the Standards' documents or related material in their schools and are in agreement with them. Significantly fewer teachers have actually made changes in their teaching consistent with the Standards or even feel prepared to explain them to colleagues. It is important to remember that "awareness" and "agreement with" the Standards is based here on teachers' self-reporting and not on objective classroom observation.

STRENGTHS

Overall the data suggests that there is an unevenness in the level of implementation. Some teachers have made changes, but many have not. However, even within those who have changed, change is not uniform nor at a level indicative of full implementation. With a recognition of this unevenness, there are some areas where progress can be reported.

At the elementary and middle grades, teachers report use of cooperative groups, an increased use of manipulatives and computers, and there is some evidence of discussion and interaction in the classroom. The greatest areas of strength at the secondary level is in the area of mathematics as reasoning, cooperative group work, and the use of calculators.

WEAKNESSES

When looking at the frequency with which most any strategy is used, it is difficult to feel complacent about the data in any given area. Clearly, there remains a lot of work to be done before we can say teachers are actually implementing the Standards.

While problem-solving is reportedly done by all teachers, the evidence does not support the use of problem-solving as a global approach to mathematics or as a pervading theme. Nonstandard and project-type problems are infrequently used. The driving force in the classroom remains the textbook.

The area of assessment is perhaps least reflective of the Standards than any other area. There is little evidence of alternative forms of assessment, portfolios, or joumals. Teachers made almost no distinctions between the use of assessment for diagnostic purposes and for grading. Traditional end-of-chapter and standardized tests remain the most common forms of assessment.

At the elementary level, calculators are used less frequently than desired and evidence would suggest that the curriculum at that level remains dominated by computational skills. While calculators are common at the upper grades and high school, there is an inadequate use of computer technology and of graphing calculators. It is not clear if this is due to lack of availability, teacher reluctance, or curricular support.

Answers to questions found in this research brief have been synthesized from the following MERC publications. Copies can be purchased using the online order form on the publications page.

Cauley, K., & VandeWalle, J. (1993, March). The NCTM Standards: Implementation survey instrument.

Cauley, K., & VandeWalle, J. (1993, November). The NCTM Standards: Implementation.

Cauley, K., VandeWalle, J., & Hoyt, W. (1993, November). The NCTM Standards: Implementation technical report.

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