Mid-continent Research for Education and Learning

The following process was used to identify standards and benchmarks for mathematics:

Identification of Significant Reports
Two basic reports were identified as the primary documents representing the current thinking on standards in mathematics: the National Council of Teachers of Mathematics' Principles and Standards for School Mathematics (NCTM, 2000) and the Mathematics Framework for the 1996 and 2000 National Assessment of Educational Progress (NAEP, n.d.). The NCTM document is a revision of the Curriculum and Evaluation Standards for School Mathematics which, published in 1989, contributed significantly to the national awareness of the benefits of identifying standards in content domains. The 1989 document was also one of the most successful standards document published in terms of the breadth of its acceptance; the process employed for the development and extensive review of NCTM's recent work suggests that Principles and Standards for School Mathematics will enjoy similar success. Additionally, even though it is not as widely known, the NAEP framework has provided strong conceptual guidance to the development of mathematics education. To prepare for the 1994 NAEP mathematics assessment, the National Assessment Governing Board awarded a contract in the fall of 1991 to the College Board to develop item specifications for the 1994 assessments. Explicit in this project was an alignment with the NCTM standards, inasmuch as they were believed to reflect the most current thinking on what students should know and be able to do in mathematics. The process of developing the recommendations for the planned 1994 mathematics assessment occurred between September, 1991 and March, 1992 and resulted in the publication of the report Framework for the 1994 National Assessment of Educational Progress Mathematics Assessment (NAEP, 1992b). Due to a budget shortfall, however, the mathematics assessments were rescheduled from 1994 to 1996 and a new framework was issued. Recently, it has been reissued as Mathematics Framework for the 1996 National Assessment of Educational Progress (NAEP, n.d.). Although the 1994 and 1996 documents were similar in many respects, there were some differences. Given that it was the more current work, and encompassed all the material found in the 1994 framework, the 1996 framework (which is also the framework for the 2000 Assessment) was used in this report.

In addition to these documents which focus solely on mathematics, Benchmarks for Science Literacy (Project 2061, 1993) contains a section entitled The Mathematical World. This section parallels and details many of the standards found in Principles and Standards for School Mathematics. Also, the New Standards Project has published three documents that report mathematics standards and benchmarks at elementary, middle, and high school levels. Those documents are: Performance Standards: English Language Arts, Mathematics, Applied Learning, Volume 1, Elementary School (1997a); Performance Standards: English Language Arts, Mathematics, Applied Learning, Volume 2, Middle School (1997b); and Performance Standards: English Language Arts, Mathematics, Applied Learning, Volume 3, High School (1997c). The Council for Basic Education has published Standards For Excellence in Education (CBE, 1998), which details mathematics content appropriate for students at the end of grades 3, 5, 8 and 12. The CBE standards are described as a "blend of state standards from Delaware, New Jersey, and Pennsylvania"(p.181).

Three documents from the International Baccalaureate Organization were also consulted: Group 5 Mathematics Guide (1993), Middle Years Programme: Mathematics (1993), and Primary Years Programme: Making it Happen in the Classroom (1996). The inclusion of the Primary Years is new with this edition.

The Third International Mathematics and Science Study (TIMSS), a large-scale, cross-national comparative study of math and science curricula, has made available to the public about two-thirds of the mathematics and science items administered to students in 1994-95. Of interest for this study were all items used for populations 1 and 2; for population 3, in keeping with the stated purpose of this study (see Process of this work), only literacy items were reviewed. The International Association for the Evaluation of Educational Achievement (IEA) published TIMSS Mathematics Items: Released Set For Population 1 (Third and Fourth Grade) (1998a); TIMSS Mathematics Items: Released Set For Population 2 (Seventh and Eighth grade) (1998b);and Released Item Set for the Final year of Secondary School: Mathematics and Science Literacy, Advanced Mathematics, and Physics (1998c).

Finally, McREL has published a study entitled A Distillation of Subject-Matter Content For the Subject-Areas of Language Arts, Mathematics, and Science (Kendall, Snyder, Schintgen, Wahlquist, & Marzano, 1999). Researchers at McREL reviewed a selected set of highly rated state standards within each subject area, examining them for common content. The McREL analysis resulted in the identification of the significant subject-area content that consistently appeared within these top rated documents.

Selection of the Reference Document
Because of its wide recognition, the NCTM document was selected as the reference report. Additionally, the report had characteristics amenable to the standards/benchmarks model used in this study. Specifically, the report explicitly identifies standards at four developmental levels-grades K-2, 3-5, 6-8, and 9-12. These new levels represent a change from the 1989 NCTM standards design, and these levels now match exactly the levels used in this study. In the previous edition of the Content Knowledge, the elements identified in the K-4 level of the NCTM document had to be reclassified into Level 1 (primary) or Level 2 (upper elementary), based on information from other documents. In this edition, the content placement for Levels 1 and 2 reflects more precisely the choices made by the authors of NCTM's newest set of standards.

Identification of Standards and Benchmarks and Integration of Information from Other Documents
The organization of benchmarks into the standards that first appeared in this study was influenced by the organization of the NAEP document more significantly than by the 1989 NCTM standards document. The NCTM standards often appeared to show little designed relationship between the content in one developmental level and that in the next. In addition, new types of knowledge and skill were sometimes introduced at a superordinate level within a standard that seemed to have no developmental relationship to the knowledge and skill identified in the subordinate level. Consequently, many elements within the various NCTM standards and levels were reclassified as more appropriately fitting within another standard. This reclassification process was highly influenced by the NAEP document. Where the NCTM document identifies 13 standards at Levels 1 and 2, and 14 standards at Level 3, the NAEP document identifies five general categories articulated at three levels roughly equivalent to the three NCTM levels. Our reclassification tended to collapse some of the NCTM standards such that the final set of nine standards (see below) resembled the NAEP classification as much as it did the NCTM classification. In effect, our reclassification tended to erode the original structure of the NCTM document.

While most of the mathematics content found in the 1989 and 2000 NCTM standards documents remains very similar, the structure of the standards has been revised. The standards themselves appear to be more deliberately articulated at all grade bands, that is, content within each standards category is addressed consistently at each grade band, whereas in the 1989 set not all standards were addressed at each grade band. In addition, a clearer delineation between mathematics content standards and what NCTM calls mathematics process standards (reasoning, representation, communication, and problem solving) has brought greater clarity to the organization of the document. For reasons discussed in the Process section of this report, many of the elements identified within the NCTM standard on mathematics as reasoning were judged to be more appropriately classified under one of the standards within our thinking and reasoning category. With the exception of the content related to problem solving, which we address in standard 1, we classify the remaining NCTM process standards as more appropriately instructional guidelines, or curriculum standards, and do not include them.

For the most part, the information in the Project 2061, New Standards, and International Baccalaureate documents was integrated into the standards generated from the NCTM and NAEP reports. The one exception to this general rule was Standard 9, Understands the General Nature and Uses of Mathematics. As the title indicates, this standard deals with general awareness about mathematics and its relationship to other disciplines, particularly science. This standard was generated solely from the Project 2061 document Benchmarks for Science Literacy.

Any benchmark that addressed content closely related to a test item in the TIMSS mathematics assessment has been identified by indicating the test level of the matching TIMSS assessment item. Similarly, any benchmark that addresses mathematics content that was also identified as important in the McREL study of top standards documents has been so identified by an asterisk at the end of the citation log, which appears just above and to the right of the benchmark.