Curriculum coherence: an examination of US mathematics and science content standards from an international perspective

Abstract

In recent years, US curriculum policy has emphasized standards‐based conceptions of curricula in mathematics and science. This paper explores the data from the Third International Mathematics and Science Study (TIMSS) to argue that the presence of content standards is not sufficient to guarantee curricula that lead to high‐quality instruction and achievement. An examination of the content topics covered in each grade of a group of six of the highest‐achieving TIMSS countries in mathematics and science shows a pattern in which new topics are gradually introduced, are a part of instruction for a few grades, and then often leave the curriculum as separate topics. This contrasts sharply with mapping of topics in the various US national standards in mathematics and science. Topics enter and linger, so that each grade typically devotes instructional attention to many more topics than is typical of the six high‐achieving countries; in addition, each topic stays in the curriculum for more grades than in the high‐achieving countries. An examination of mathematics and science content standards from 21 states and 50 districts in the US shows a pattern more like that of the US national standards than those of the high‐achieving TIMSS countries. While content standards have become integral to US curriculum development and reform, they have yet to reflect the coherence that is typical of countries that achieved significantly better than the US in the TIMSS study.

Keywords:

• mathematics
• curriculum
• science curriculum
• curriculum content
• standards
• academic achievement

Notes

1. The term ‘math wars’ reflects a distinction made in the US between two points of view regarding the nature of school mathematics: the mathematics curriculum needs to focus more heavily on learning basic facts and mastering standard algorithms; the mathematics curriculum needs a greater focus on the process of mathematics and problem‐solving more generally. This has also been described as ‘traditional’ vs ‘reform’ mathematics.

2. A review of these and other definitions of coherence can be found in Newmann et al. (2001).

3. The 1989 NCTM Standards were updated in 2000 (NCTM 2000).

4. Throughout this paper, the US national standards are derived from the standards documents produced by national professional organizations such as the American Association for the Advancement of Science (AAAS), NCTM, or the NRC.

5. Several caveats are necessary to characterize the procedures more carefully. The first caveat derives from the situations where the next topic in the order was tied with at least one other topic in terms of the percentage of A+ countries covering it. In such cases, all tied topics were listed as optional, with an indication of the number of topics to be selected from the optional list. Secondly, if less than the majority intended the next set of topics in the descending order, and more topics were necessary to reach the criterion level for that grade level, then these were listed as optional. Thus, the ‘optional topics’ can be thought of as those topics where choice is possible in terms of rounding out the benchmark topics to the typical number of such topics intended for instruction at that grade level by the A+ countries.

6. We (see Schmidt et al. 1997b) divided the content standards into small segments called blocks. After defining the blocks, the actual instructional material in each block was described using categories from the TIMSS curriculum frameworks, i.e. coders identified each block’s content in terms of the topic(s) involved (44 different topic codes for mathematics, 79 for science). More complex standards can be identified with more than one topic as appropriate. Coders (graduate students with degrees in mathematics, engineering, and the various sciences) were provided with training to master the techniques. These coders produced a detailed line‐by‐line, page‐by‐page content analysis of the standards using the framework content codes. These content codes also provided the basis for the GTTM procedure used in the development of the international benchmarks. The GTTM procedures not only used the same codes, but involved the same type of documents from each of the participating countries. However, those procedures did not involve the more formal document coding but only their use by experts to make informed judgements. Training materials relevant to these techniques are found in the SMSO Technical Report Series No. 2. http://ustimms.msu.edu/techrep/techrep2.pdf (visited 4 August 2004).

7. Note that both of these aspects of curriculum design, although possibly influenced by developmental and motivational issues, seem more closely aligned with the nature of the discipline than with such issues. For example, in the US it is sometimes suggested that certain topics contained in figure 1, such as functions, cannot be covered in the middle grades because the students are not developmentally ready. Some topics are intended to be introduced early in the US curriculum because it is (supposedly) good for motivational purposes or develops the groundwork for its own later coverage, even though this might be inconsistent with the mathematics itself, e.g. congruence and similarity. We conjecture that the topics found in the bottom half of figure 1 were not covered until the middle and upper grades, primarily because the pre‐requisite knowledge has to be covered and mastered first—and not primarily because of other issues. Such issues can clearly not be ignored, but they should assume a less dominant role in the sequencing of topics than often appears to be the case, at least in terms of some proposed US curricula.

8. These included arithmetic topics such as whole numbers; measurement topics, including units as well as perimeter, area, and volume; data analysis; equations; and 2‐dimensional geometry, including points, lines, angles, and circles.

9. Of course, doing this alters the basic conception upon which figure 1 was based, the idea that a majority of the A+ countries intended coverage of each of these topics. However, it is an attempt to represent a composite curriculum typical of the TIMSS top‐achieving countries. However, this conservative approach introduces another problem: it suggests more topics than would be typical for the A+ countries. However, the inherent ambiguity permits few if any reasonable alternatives.

10. We used the 1989 edition of the NCTM Standards since these were the ones in place at the time of TIMSS. Given the timing, they would also have influenced the state and district standards analysed in the next sections. We also analysed the 2000 version of the NCTM Standards (NCTM 2000). The grade‐level structure of these new standards is different. The grades are grouped as follows: grades 1–2, 3–4, 5–6, and 7–8.

11. The 21 states included California, Colorado, Delaware, Florida, Georgia, Illinois, Indiana, Kentucky, Maryland, Massachusetts, Michigan, New Hampshire, New Jersey, New York, North Carolina, Ohio, Oregon, South Carolina, Vermont, Washington, and Wisconsin.

12. This is not surprising since the NCTM Standards were used as a model in the development of many state standards (Blank et al. 1997).

13. The topic as listed in the original TIMSS framework was slope and trigonometry, but for most countries this was mainly about slope (see Schmidt et al. 2001).

14. We analysed the sequence of intended coverage for each of the state mathematics standards separately in order to determine if figure 4, using a composite of the 21 states, distorts the reality of individual state standards. This was not the case. In fact, for many of the individual states, we find that almost all topics are intended to be taught to all students at all grades.

15. The comment contained in note 9 is especially relevant to science.

16. The patterns of the 21 individual states are mostly similar to the national standards, and quite different from the upper triangular pattern of the composite curriculum of the top‐achieving countries. For the 41 science topics (out of the total of 79 topics contained in the science framework), all of these states, with only three exceptions, have many topics introduced at grade 1 that last through grade 8. The most obvious example is the standards for one state where 41 topics are intended for coverage starting in grade 1.

17. Valverde et al. (2002) provides data on the diffuse nature of US textbooks.