The NCTM Standards: Recommendations for ActionMetropolitan Educational Research Consortium

MERC'S study of the implementation of the National Council of Teachers of Matematics Standards provides data to corroborate the viewpoint that much work needs to be done if school systems are to improve mathematics programs in elementary and secondary schools. The recommendations for action are provided to foster a continuing dialogue at the school system level to implement the Standards.

CURRICULUM AND INSTRUCTIONAL PROCEDURES

Elementary Level

  1. The entire area of computation at the elementary level should be looked at. The data suggest that most teachers spend the majority of their time on pencil-and-paper computational skills while spending little time on mental computation and computational estimation. It is fairly clear that the current emphasis is a function of textbooks, standardized testing, and long-standing traditions, each of which have consistently stressed pencil algorithms as the backbone of the elementary mathematics curriculum.

    The Standards calls for a de-emphasis on these outdated skills with an increased emphasis on more flexible and more frequently used mental methods. Teachers will need inservice and support from curriculum materials to make this change.

  2. While teachers report addressing the area of problem-solving, it is not clear that the full curriculum is being approached in a problem-solving manner. The first theme standard is mathematics as problem-solving. To implement this theme requires teachers to have a more complete understanding of the full intent and philosophy of the Standards. Teachers still seem to see mathematics as a rule-driven curriculum rather than one in which students are involved in the discovery and invention of mathematical ideas through problem-solving methods.

  3. The data suggest that calculators are only used infrequently in the elementary school. Based on the focus group interviews and on informal observations, it is most likely that very few teachers require students to have calculators or have them readily available at all times. The Standards calls for calculators to be available at all times and to be used in all areas of mathematics including assessment. There are a number of ways to get inexpensive calculators into the hands of every elementary child, including making them required material for school. The daily or regular use of calculators must begin with availability and be followed with ideas for using them.

  4. The areas of geometry, probability and statistics should be given significantly more visibility in the required curriculum. The evidence suggests that geometry receives only minimal attention and that probability and statistics are rarely taught. Once again, textbooks, testing programs, and lack of a tradition of teaching in these areas are the targets that need to be addressed.

  5. While primary grade teachers are comfortable at least with the idea of using manipulative materials, upper-grade teachers are facing unfamiliar challenges. They lack experience with managing manipulatives in the classroom. They do not have experiences with making materials nor do they have adequate commercial supplies to support a manipulative approach. They also need help with ways to use manipulatives with older students.

Middle School Level

  1. As at the elementary level, computation and number sense appear to receive the most attention in the middle-grades curriculum. Although it may appear from the data that teachers include number sense in their instruction, it is not clear that the term number sense is clearly understood in the same sense as is meant by the Standards. At this level, mental computation and computational estimation are clearly a component of number sense as is a connection of number concepts with real word referents. In these areas, teachers are not reporting strong instructional emphasis. More attention should be given to a broad view of number sense while simultaneously deemphasizing pencil-and-paper computation. Textbooks, testing, and teacher knowledge of the curriculum must all be addressed in order to promote change.

  2. Middle school teachers seem to be considerably more textbook bound than their counterparts at the elementary level. They need more assistance in how to teach mathematics through open-ended explorations, projects, and group work. A problem-solving approach and the use of manipulatives in middle grades is rarely seen. A more exploratory, discussion-oriented approach is recommended by the Standards.

  3. The middle school teachers tend to use calculators more than at the elementary level but not on a daily or even regular basis by any means. The use of computers is actually weaker than at the elementary level. In fact, due to some of the more recently available computer software, the teacher of the middle grades has more reason to use computers now than ever and the calculator should be an ever-present tool at the disposal of the middle-grades student. In fact, in the 1992 NCTM Yearbook on calculators in mathematics, a strong case is made for the use of graphing calculators at the middle grade level.

  4. The curricular areas of patterns, measurement, statistics and probability are all in need of increased emphasis in the middle grades if the Standards are to be addressed.

Secondary Level

  1. The notion of a core-curriculum at the secondary level is one of the main themes of the NCTM Standards. The core-curriculum concept envisions all students having access to significant mathematics with no one being denied access due to lack of computational skills. The differentiation between college-intending and noncollege-intending students is determined by the depth of study, not what is studied. Thus, all students should have access to algebra, geometry, probability and statistics, and discrete math topics.

    The current study was not designed to determine specifically if a core curriculum was in place. However, there was little evidence that suggested the existence of a core curriculum. For example, there remains a lot of traditional instruction in skill areas for lower-level students. No discussion of addressing a variety of mathematical topics for all students was noted. To move toward a core curriculum is a major step for any school division and requires significant changes in the total curricular offerings for the four years of secondary schools. While the study itself did not uncover such changes, it is noted that at least two of the MERC school divisions have begun implementation of some form of a core-curriculum concept.

  2. The secondary teachers in the study report using calculators fairly regularly. However, it is clear from the focus groups that graphing calculators are far from a standard tool. It is more likely that only one or two teachers are using graphing calculators or have taken the time to learn how to use them. For under $70 each, these calculators are essentially small computers that can be programmed, used for investigating graphs, working with statistics, computing matrices, and much more. Students in college-bound programs should be encouraged to purchase their own graphing calculators, schools should have them available for use, and most importantly, nearly every secondary school course should take advantage of them. This will require some training or at the very least some support in terms of curricular materials.

  3. Computer usage is also weak and spotty. In the opinion of the teachers, much of the problem is hardware and software compatibility. Simply purchasing computers and/or software is not sufficient Teachers must be afforded time and support to learn about the software and computers must be installed in usable configurations and be compatible with the desired software.

  4. Secondary teachers could improve their problem-solving approach to instruction including the use of cooperative learning groups, project problems, and more student writing.

ASSESSMENT

Of the areas investigated, assessment strategies and procedures was the area where teachers have made the least amount of change in the direction of the Standards. This is true at all grade levels and little is gained by separating the three levels in this discussion.

The predominant mode of assessment is the chapter test and related quizzes. While teachers report that they assess concepts as well as procedures, an examination of the typical chapter test will indicate that the conceptual understanding required is minimal. Most teachers are concerned about standardized testing, reporting numeric grades backed up by test averages and the performance of routine procedures. While these are not bad objectives, they represent an incomplete approach to assessment.

In their reporting, teachers made almost no distinction between the use of any assessment procedure for grading purposes versus instructional feedback or diagnosis. One interpretation that may be made is that they really do not make this distinction at all. It is unlikely that they do as much assessment for diagnosis as for grading as is suggested by the data.

The Standards calls for assessment to be much less distinct from instruction than has traditionally been the case. By broadening an assessment plan to include observations, checklists, portfolios of work, group projects, and performance tasks, the things that students do in class as part of their learning experiences can also be included in an assessment plan. Furthermore, the Standards calls for alignment of all assessment, not just with objectives of the course but also with the methods of instruction. If calculators and/or manipulative materials are used in the instructional program perhaps they should also be used in the assessments.

With the possible exception of some observations and group work being used at the elementary level and calculators permitted in most secondary class testing situations, it is safe to say that teachers are still using the same assessment procedures that they always have.

The researchers' experiences with teachers taking courses suggest that learning to implement and use effectively a broader, performance-based plan of assessment that is integral to instruction, is a very difficult task. It requires considerable instruction in assessment strategies and having the opportunity to try them out in the classroom. Teachers must begin small and find one or two new ideas that suit their personal style and agenda. From this beginning they can gain confidence and add additional strategies later.

Teachers who do adopt a variety of alternative assessment procedures tend to be very positive about the results – except for the hard work that is almost always involved. If school divisions want to help teachers look more broadly at student achievement and report more accurately to parents what students are able to do (instead of what they cannot do), schools must do more than provide one-shot inservice. They must develop a plan that will guide and support teachers, offer them choices, and show them the benefits of the additional work involved. This is easily one of the most difficult areas of the Standards to implement.

AIDS TO IMPLEMENTATION

The data, especially the focus group interviews, provide useful information concerning the types of things that can influence (or hinder) change and general implementation of the Standards. Some of these things must involve division change or division implementation. We might call these structural changes - changes that are beyond the control of the individual teacher or principal. Other factors influencing change are more local. These factors are under the control of the building principal or department chair, or can be dealt with through inservice and other avenues of professional growth. Local factors are still influenced at the division level. For example, providing quality, focused inservice is listed here as a nonstructural aid. However, to follow up on that inservice or to implement the suggestions remain in the control of the teachers and principal. Implementation of a core curriculum at the secondary level is clearly structural, requiring division implementation.

Structural Initiatives

  1. The development of a clear policy relative to the NCTM Standards, coupled with information and appropriate modifications in the curriculum, would help teachers and principals have a sense of direction that is sometimes lacking. Teachers, even those who are well aware of the Standards, are not clear about what they themselves should be doing in the classroom. Conflicting messages concerning objective lists, SOL’s, ITBS and other mandates are not always in sync with the message of the Standards. It is not reasonable for teachers to be expected to make change with only a simple admonition that we support the Standards. Information about the Standards and how the division views implementation is important for all concerned.

  2. Testing policies should be examined to be in keeping with the Standards. It is, of course, important to be aware of test scores as a measure of how well a school or division is doing. However, it is now very important to take a careful look at the items that are included in those tests. The current version of the ITBS is heavily weighted toward procedural or algorithmic knowledge with very little emphasis given to concepts and problem-solving skills. Textbooks also have chapter-end tests that tend to focus on the lowest level skills of the chapter.

  3. Examine policies relative to textbook adoption and be certain that they are in keeping with the spirit of the Standards. When textbooks are not in keeping with pronounced directions, either because they are out of date or because they were selected with out-dated criteria, they are of no real help to teachers.

Local Initiatives

  1. Promote and support the lead teacher concept. Many teachers talked of the value of having a lead teacher in the school – someone they could count on to have current information or a good idea to solve a problem. Lead teachers are present in many divisions but require the constant support of principals since no moneys are available for mathematics specialists. At the secondary level, the department chairs should be encouraged to be instructional leaders in the same way as the lead teacher is at the elementary and middle school level.

  2. Make opportunities for teachers to observe one another within the buildings and encourage teachers to share ideas and problems about teaching mathematics. Teachers truly value ideas that come from their colleagues.

  3. Provide quality inservice that is focused on specific classroom issues in the teaching of mathematics. Teachers react negatively to inservice that is not specifically useful to them or is so general in nature that significant implementation problems make using the information nearly impossible. Relevance of inservice is very important. If the inservice is good, support should be available to follow up on it. That support should come in the form of sharing, materials, encouragement, and time to work on implementation.

  4. Principals need to be knowledgeable about the NCTM Standards. While a relatively high number report awareness of the Standards, there is also evidence that a principal who is truly involved in making the Standards a reality in the school is somewhat of an anomaly. As teachers spend extra time and effort at implementation of new ideas and request support for materials, manuals, software, calculators, and help from others, these efforts and requests must not fall on deaf ears. Teachers cannot implement the Standards without support.

  5. Be very sure that teachers are aware of opportunities for professional growth. Professional journals and books from NCTM could be made available. Opportunities to attend conferences at the local, state, regional and even national level should at the very least be well publicized and discussed. When possible, funds, such as Eisenhower money, should be made available to get teachers to these meetings. Teachers should be encouraged to attend special training courses, take workshops and even university courses. When teachers do make these efforts, their efforts should be rewarded - at the very least with recognition and praise if not with more tangible means of support for work in the classroom.

  6. Examine issues and concerns around technology. Simple calculators should be available to every child at all times. Teachers should not have to go to a central place to get the "grade-level" set. Where reasonable, school policy could permit students be required to bring a simple calculator to school. For more expensive technologies, similar policies should be made in cooperation with the teachers. Care should be taken to see that computers that are available are in good repair, that there is reasonable software to use on the computers. All teachers at all levels need help in how to use technologies in their courses. It is not reasonable to expect them to use a new tool without assistance.

CONCLUSION

The NCTM Standards document is now five years old and the Professional Teaching Standards are three years old. These recommendations, while universally accepted across the nation, are not likely to be easily implemented in their entirety. NCTM itself recognizes that true implementation is a long-term endeavor that will extend into the next century. The findings of this survey are generally consistent with these expectations. There is some movement in a positive direction as more and more teachers are at least aware of an agenda for change. Actual significant change on any global basis, however, is very difficult to find. Teachers are in serious need of support as well as education concerning what the Standards are actually saying.

Glenda Lappan, Chair of the Commission on Teaching Standards for School Mathematics, writes:

The kinds of change called for by the vision in the standards documents are so fundamental and pervasive that they seep into every aspect of our society. The current curriculum, expectations, and teaching practices are failing with so many of our students in mathematics that we have a responsibility to rethink what we are about. To accomplish change on a large scale, all the stakeholders - students, teachers, parents, school administrators, business, industry, professional mathematicians, politicians, and others - need to understand the issues and the direction of reform and give their support to the effort.

Information found in this research brief has 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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