Over the past two decades, there has been an increasing demand for quality public education emerging from many avenues -- concerned parents, the business community, governors, academics, economists and politicians, to name some of the more vocal (A Nation at Risk, 1983). This has fostered the standards-based education movement (Ravitch, 2000). The basic notion undergirding this movement is that an educational system should be guided by content standards defining what it is that students should be expected to know or do. The concept essentially is practiced in most countries, at the least in the almost 50 that participated in the Third International Mathematics and Science Study (TIMSS) (Schmidt, McKnight, Valverde, Houang & Wiley, 1997; Schmidt, Raizen, Britton, Bianchi & Wolfe, 1997). In the vast majority of these countries national content standards are present that emanate from a national institutional center responsible for curriculum policy (Schmidt et al. 2001). In effect most of these countries use content standards as a major purveyor of policy defining the vision of what is important for children to learn as a part of their schooling. Standards-based school reform in the US attempts to use content standards in the same way.
There are two difficulties in applying this widely used practice in the American educational system ñ the definition of what constitutes an educational system and identifying the institutional center that is to set the curricular policy. The US has a notion of shared responsibility in curriculum decision making and a complex structural arrangement for schooling involving over 15,000 local school districts. This makes what is relatively simple and broadly used internationally much more difficult to operationalize. It also makes it much more political. What many other countries take for granted is problematic in the United States.
The different sides of the issue center on beliefs about the concepts of ëAmerican individualismí and ëlocal controlí (Timar et al., 1998). Most voices against the use of educational standards, or at least standards set at a national or state level, have been focused on respecting local differences in childrenís needs and allowing them to play out at least at the district (if not the individual school) level. The fundamental argument seems to be that, while it may be desirable to have children within the country or a state study according to the same content standards, the desirability of this uniformity is outweighed by the undesirable nature of having such standards imposed from beyond a local district.
Nevertheless, over the years consensus on setting educational standards to reform public education has gradually emerged. However, as a result of the battle pertaining to respecting local choice the US has a plethora of standards defined at each of the different levels ó what we have argued elsewhere results in a "splintered vision" (Schmidt, McKnight & Raizen, 1997). Most students attend schools which have at least state if not both district and state content standards (Education Week, 2000). No mandatory national standards exist in the US but rather only non-binding recommendations. Such standards as exist were developed not as official USpolicy but as recommended policy. They were developed not by the federal government but by national organizations such as the National Council of Teachers of Mathematics (NCTM) and the National Research Council (NRC) associated with the National Academy of Sciences.
All of
this seems to suggest that the US has overcome its reticence
regarding the use of content standards and the difficulty of applying
the concept in an educational system that is really not one but
something more like 15,000 different systems. (Judge et al.,
1994) The political solution has been to allow standards to emerge at
different levels in the system. The political and public reaction
seems to suggest that as long as there are standards of some sort, at
some level, this is all that is necessary and, presumably, sufficient
to attain needed reforms. This has led to various debates about
content standards that were often not attached to particular
disciplinary substance. This has generated confusion rather than
focusing the policy discussion on the particulars of the relevant
subject matter. Hayes Mizell captures this confusion well in the
following remarks:
Standards, standards, standards. Is there another word that sparks so many different images and reactions? Some people believe standards mean testing. Others believe standards mean accountability. Many people think standards are a force powerful enough to shape the content and method of instruction. Still others view standards as a kind of wolf in sheepís clothing that justifies grade-level retentions and pushing students out of school. There are even some people who believe that standards, in and of themselves, represent the magic key to school reform that will unlock the mysteries of teaching and learning. (Remarks made on October 20, 2001 at the Conference on Teaching and Learning sponsored by the Association for Supervision and Curriculum Development.)
We would like to argue here that the
important issue is not merely whether there are content standards,
but the quality of those standards. Concern over quality has more
recently emerged as an issue and evaluations of state standards have
become available within recent years (e.g. Thomas B. Fordham
Foundation, 2000). These assessments have been based on
characteristics related to clarity, specificity and, often, a
particular ideology from which it is desirable that subject matter
definition should derive. For example, in mathematics these
distinctions have been revealed in what is called the 'math wars,' a
debate over what constitutes basic mathematics for the school
curriculum. Our definition of quality moves beyond the above issues
to what we believe is a deeper, more fundamental characteristic. We
feel that one of the most important characteristics defining quality
in content standards is what we term coherence. Further we argue that
coherence itself implies the necessity, at least at some level, of
uniformity across schools or districts especially given the US
practice of shared decision making.
The concept of coherence is used in different ways in the literature. One term used to describe a particular type of coherence is alignment, by which is meant the degree to which various policy instruments available to the system such as standards, textbooks and assessments accord with each other and with school practice. (Consortium for Policy Research in Education, 2000; Fuhrman, 1993; Smith & OíDay, 1990). Another definition of coherence focuses on school organization. Issues of organizational focus, an articulated vision and a common culture of values become important in defining a coherent system (Bryk, Lee & Holland, 1993; Celio, 1998; Coleman, Hofer & Kilgore, 1982). An excellent review of these and other definitions of coherence can be found in Newmann, Smith, Allensworth & Bryk (2001). In that same article the authors define instructional program coherence along the lines of what some have called alignment -- a set of interrelated programs for students and staff that are guided by a common framework for curriculum (Newmann, Smith, Allensworth & Bryk, 2002, p. 297). Their actual definition is more complete and complex but the essential point here is the reference to a common curriculum framework. In fact, the authors suggest that instructional program coherence entails curricular coherence by which they mean (p. 298) 'sensible connections and coordination between the topics that students study in each subject within a grade and as they advance though the grades.'
It is this latter definition that comes closest to what we mean when we speak of coherence as a central defining element of high quality standards. If one of the major purposes of schooling is to help students develop an understanding of the various subject matters deemed important by a society, such as mathematics and science, then the definition of ëunderstandingí is important to examine as a way of viewing each discipline intended for schooling.
Bruner (1995, p. 333) suggests that ëto understand something well is to sense wherein it is simple, wherein it is an instance of a simpler, general case. In the main, however, to understand something is to sense the simpler structure that underlies a range of instances, and this is notably true in mathematics.í Brunerís definition implies that the structure of the content in a discipline is important and that, for example, the goal of helping students understand mathematics is facilitated by making visible to them an emerging and progressive sense of its inherent structure. Bruner (1995, p. 334) describes this as ëopt[ing] for depth and continuity in our teaching rather than coverage . . .to give him (the student) the experience of going from a primitive and weak grasp of some subject to a stage in which he has a more refined and powerful grasp of it.í
We define content standards in the aggregate to be coherent if they are articulated over time as a sequence of topics and performances that reflect a logical and, where appropriate, a sequential or hierarchical nature of the disciplinary content from which the subject matter derives. This implies that a set of content standards ëto be coherentí must evolve from particulars (e.g., simple math facts and routine computational procedures associated with whole numbers and fractions) to deeper structures. It is these deeper structures by which the particulars are connected (such as an understanding of the rational number system and its properties). This evolution should occur both over time within a particular grade level and as the student progresses across grades.
Content standards that are not based on the progressive structure of the discipline seem likely to appear arbitrary and to look like a ëlaundry listí of topics. Of course, in an area that is largely hierarchical (such as mathematics) even a ëlaundry listí approach will usually not violate the more obvious hierarchical aspects of the discipline, e.g., by placing the coverage of percentages before the coverage of common fractions. However, subtler aspects in which this lack of coherence might manifest itself is in the ëintroductioní of a topic before prerequisite knowledge is covered that makes a reasonable understanding of the topic possible.
For other disciplines such as science (at least as portrayed in school science), where knowledge is not quite as hierarchical, and where multiple disciplines are involved (such as biology, physics, chemistry and geology), the temptation for the ëlaundry listí approach to content standards is much greater. This is especially the case in terms of cross-disciplinary coverage where the sequential interleaving of topics from the multiple science disciplines may be quite arbitrary.
The definition of coherence that we propose presupposes that the standards move progressively towards the understanding of deeper structure. How deeply they go into that structure reflects decisions about what all functioning members of the society should know versus what those who will specialize in fields related to these disciplines need to know. For the example from mathematics involving fractions and whole numbers, a deeper structure from the rational number system might include basic knowledge of rings, fields and groups. However, in general, this has not been viewed as essential knowledge for most members of the society not involved in mathematically related occupations or fields of study.
How deeply into the structure of the discipline one moves and by what grade level one moves to that depth represents an aspect of coherence ó which we term 'rigor'. This idea of increasing rigor seems implied by the progressive nature of understanding inherent in Brunerís definition. This definition of rigor might appear to be arbitrary and perhaps unimportant. What does it matter to the concept of understanding at what grade level it takes place? However, when it is coupled with the fact that at higher grades not all students will necessarily come under the same set of standards, rigor has clear implications in terms of facilitating an understanding of that content which has been deemed fundamental for all. Suppose, for example, that the desired rigor of a particular set of standards in eighth grade is first attained in another set of standards at eleventh grade and the level of understanding of the discipline implied by these first standards is essential for all students to acquire. If so, serious issues of access and equity arise in the latter case given the practices of self-selection and tracking prevalent in US high schools.
At a simpler level, it seems to us that if content standards reflect a logical structure of a discipline, then these standards should progress in terms of depth and rigor as one moves from elementary to middle school. This concept of coherence implies replacing the current US situation of almost endless repetition of the same standards across grade levels with a progression of standards over those same grade levels. Such a progression would represent a continuing penetration of the discipline moving to a deeper structure that makes things ësimplerí in Brunerís terms. (Schmidt, McKnight & Raizen, 1997; Schmidt, McKnight, Cogan, Jakwerth & Houang, 1999).
A second consequence of our argument in the US is the definition of coherence that we have advocated seems to imply the necessity of at least some degree of uniformity across the myriad of systems that define US schooling, if not for the US as a whole. We have argued that for content standards in the aggregate to be coherent requires that they must be articulated as a sequence of topics and performances that reflect a logical structure of the corresponding discipline. This, it seems to us, is particularly difficult to achieve in the US context of shared decision making where the primary driving forces in the development of standards are political compromise, ideological concerns and the need to reflect current practice -- that is, what is currently being taught in classrooms.
These threats to coherence which can be present in any educational system have exacerbated impact in a political context where shared decision making leads to multiple different visions and in which those multiple visions must be commonly supported by nationally developed textbooks and standardized tests. The interplay of these factors in this context seems certain to result in standards that are more like ëlaundry lists.í That is, the standards that result are likely to be sequences of topics and performances based on some kind of composite notion, arrived at by compromise, that is all-inclusive, and that is reflective of current practice at least as represented by the most commonly used textbooks. The all-inclusive characteristic reflects the fact that it is politically safer to include topics than to exclude them, especially those over which there is controversy. Political compromise as an organizing principle for standards seems unlikely to be consistent with the principle of coherence.
This has led at the national level to ëno simple, coherent, intellectually profound and systemically powerful visions guiding US mathematics and science educationí (Schmidt, McKnight & Raizen, 1997, p. 89). Why would this, however, make it more difficult for an individual state or district to develop coherent standards? The reason lies in the complexities of the resulting interactions involving states, districts, professional associations, teachers and textbook and test publishers.
The development of standards even at a very localized level does not occur in a vacuum in spite of rhetoric to the contrary used in defending local school control. These processes are influenced by standards from other organizations such as other districts, states and national associations. In addition, these discussions involve examinations of textbooks and standardized tests as well as an intuitive sense of what is being taught in the classroom itself. The resulting multiple possibilities coupled with the American cultural notion of individualism and the virtual absence of input from the academy (university professors, such as research mathematicians and scientists) make defining the sequence of topics more an exercise in democratic consensus making. Certainly it is less the result of an intellectual exercise reflecting the logical structure of the discipline. Further to have these discussions take place together with the necessity for political compromise increases the likelihood of ad hoc approaches to content rather than adherence to underlying logical structures. All of this is made more difficult experientially by the presence of some 15,000 different settings in which this scenario must occur. Put simply, it is not surprising that the result of such a political process are standards that are more like laundry lists than reflective of the inherent logic of the disciplines on which they are based.
The
resulting lack of coherence most likely the result of the
standard-setting process in the US is understandable from an
organizational point of view if we view the standard-setting process
as being defined by the ëorganizational process modelí of
Simon (1975). This is reflective of an earlier discussion of Simon's
model:
Formally, this situation is a loose coupling of several relatively independent ëactorsí. What is the basis for their actions and decisions and how does this affect the aggregate coherence of emerging policies? One possible organizational model is a classical, ërational actorí model. This model considers each actor to behave rationally in making individual decisions with an eye to the cumulative effect of those decisions on the whole. The strong mutual concern and sense of shared responsibility implicit in this model does not seem characteristic of our situation today.
An alternative model seems likely to be more germane. This is an ëorganizational process modelí that views government as a conglomerate of many loosely allied subunits each with a substantial life of its own. This certainly seems more characteristic of the loose federalism especially true when we include secondary actors, such as professional organizations and textbook and testing organizations, in the picture. Each ëactorí pursues his or her own ëlifeí -- his or her goals, visions, plans, processes, and efforts to satisfy those to whom he or she is accountable. The aggregate effect of these separate lives is a secondary concern for most.
The decisions, policies, and documents that flow from this conglomerate of ësub unitsí should be considered not so much as deliberate choices contributing to an aggregate effort. Rather, we may better consider them ëoutputs of large organizations functioning according to standard patterns of behavior.í Thus, the parts of our federalism not only act with primary reference to their own internal life, but they operate by traditional patterns. These patterns vary in how integrally they include attempts at ërationalí decision-making and concerns for aggregate effects beyond their own particular concerns. (Schmidt, McKnight & Raizen, 1997, p. 92)
All of this points towards uniformity, at least at some level of aggregation, as likely necessary to achieve coherent standards given the structure of the US educational system.
This article presents data related to the
coherence and rigor of content standards, contrasting those of
several high achieving TIMSS countries with a selection of those from
US national professional organizations, states, and districts.
In 1989, the release of the Curriculum
and Evaluation Standards for School Mathematics by the National
Council of Teachers of Mathematics crystallized the idea of
standards-based educational reform in mathematics. It also led to an
intensive process of generating standards documents in other subject
matter areas, including history, language arts, science, and
geography. The participating parties included professional education
organizations such as the American Association for the Advancement of
Science, which produced the Benchmarks for Scientific Literacy
(AAAS, 1993), the National Research Council which produced national
science standards and most state departments of education. Education
Week's (2000) annual report, Quality Counts, reported that 44
states had standards in all four-core subjects: English, mathematics,
science, and social science. Except in Iowa, every state's department
of education organized and engaged educators to produce a state-wide
educational standards document in at least one core subject in
kindergarten through twelfth grade (Education Week, 2000). School
districts have also played a role in this large-scale production of
content standards. District-wide standards have been developed in
many local school districts across the country. Consequently, many
schools have been confronted with multiple national, state and
district content standards as they decide what to teach their
students.Smith and O'Day (1990) indicate that establishing learning
standards is only one piece of the puzzle in systemic educational
reform. Although it may be a tiny piece in the reform landscape,
setting educational standards has been a challenge for Americans
(Ravitch, 1995; Resnick, N. & Resnick, 1995). As previously
mentioned, an argument over American individualism and respect for
local choices is constantly brought forward to argue against
broad-based standard setting, especially at the state or national
level. Meanwhile, textbook and test publishers have implicitly taken
over the task of setting broad standards for American students. This
results because, textbooks and tests that are nationally designed,
written and sold become the primary common content element in US
education when there are no agreed-upon, broad-based standards.
The lack of consensus in defining priority content in this country has inevitably resulted in unfocused curricula that are reflected in most statesí content standards and in the textbooks that teachers use. The splintered and sporadic content commonly experienced by students in the US is specified through these state standards (Schmidt, McKnight & Raizen, 1997). Without coherence in content standards, a ëmile-wide and an inch-deepí curriculum is specified and seems certain to be one of the critical factors contributing to Americaís poor performance in TIMSS. (Schmidt et. al, 2001)When TIMSSí achievement results were first released, there were those who expressed concerns that the poor US performance would be used to negatively assess the effectiveness of standards-based educational reform. They argued that change would not occur quickly and thus the poor performance in TIMSS should not be used to criticize the standards-based reform movement. However, recently, TIMSS-Repeat (TIMSS-R) results have been released and students have still not shown significant progress (Pursuing Excellence, 2000). The TIMSS-R eighth grade students were chosen from the same cohort in TIMSS who were in fourth grade in 1995. This was the same cohort of students who outperformed all but one nation in science in 1995 (Pursuing Excellence, 1997).
Should these latest results be used to again raise the question of whether standards-based reform ought to be done away with as a result of this lack of improvement by American students? We believe the answer to this question is ënoí but we also believe that the more relevant question is what characteristics of US content standards are different from those of the rest of the ëTIMSS worldí and contribute to the USís lackluster performance. We hypothesize that one such characteristic is the lack of coherence.
The TIMSS curriculum study showed the US mathematics and science curriculum to be unfocused, repetitive and to be undemanding by international standards (Schmidt, McKnight, Cogan, Jakwerth & Houang, 1999). So TIMSS results, in fact, create a very clear and powerful argument in favor of standards-based education. However, they change the focus of the dialogue to one of how those standards can be improved. Abandoning the idea of content standards will not improve students learning. To some in the US the issue seems to be one of ëhaving or not having standardsí as if having them were enough. TIMSS results suggest that the intended content (the topic and performance expectations), how it is sequenced, and how demanding it is are the critical issues. Content standards, if they are incoherent and aimed too low will likely be related to students falling short on international mathematics and science achievement tests. Thus, the issue is of content standards' coherence, not merely of their presence.
Challenging as it may be to define content standards in this country, a commitment to establishing standards that are coherent is critical to educational reform (Schmidt et al, 2001). More than a decade has passed after the release of the initial NCTM mathematics standards. Numerous standards documents have subsequently been produced in various venues -- professional organizations, states, and school districts. Thus an urgent need arises to examine the coherence of these standards. To provide a context in which to view these various US standards we use data from TIMSS characterizing the content standards of the top achieving countries.
A methodology was developed to define the commonalties in subject matter expectations in mathematics and science across the top achieving TIMSS countries. Using the resulting patterns of intended coverage as an international benchmark for ëqualityí in mathematics and science standards, we look at issues of coherence and rigor in the composite standards for the top-achieving countries in TIMSS and those of 20 states and 60 school districts.
The Third International Mathematics and
Science Study (TIMSS) is the most extensive and far-reaching
cross-national comparative study ever attempted (Beaton, Martin,
Mullis, Gonzalez, Smith & Kelley 1996; Schmidt & McKnight
1995; Schmidt, et al, 2001). TIMSS tested three student
populations. This included Population One, students who were mostly
nine years old (grade three and four in U.S.. It also included
Population Two, students who were mostly thirteen years old (grade
seven and eight in U.S.), and Population Three, students in the last
year of secondary school (twelfth grade in the US). US fourth graders
performed relatively better in mathematics than the eighth graders
and eighth graders performed relatively better than the twelfth
graders.
Discussion of the TIMSS achievement results have prompted US policy makers, as well as those from other countries, to consider more carefully the curriculum portraits TIMSS has produced, especially those for the highest achieving countries. (Schmidt, McKnight, Valverde, Houang & Wiley 1997; Schmidt, Raizen, Britton, Bianchi & Wolfe, 1997) This has led to an effort to discern just what it might mean to have a ëworld classí mathematics or science curriculum (Valverde & Schmidt, 2000).
Looking beyond one's own educational practices makes a critical examination of long-established beliefs and practices in education possible. It allows individuals to reflect on the ways educational policy choices are made and to examine the resulting consequences for students. Eisner (1993) used hypothetical educational ideologies to describe the beliefs, values, or visions that ultimately frame the curriculum of an educational system. Of all the educational ideologies described, the academic excellence ideology is how we have defined the international comparison group in this article. We investigated the top achieving TIMSS countries' mathematics and science content standards to distill what they considered essential content as elucidated through an amalgam of their mathematics and science content standards over the different grades of schooling. TIMSS included comparisons of the content standards of approximately 50 countries.
We use a definition consistent with that laid out in Valverde & Schmidt (2000) which examined the content standards in mathematics and science of the TIMSS top achieving countries. We devised a methodology to determine the elements that were common to these countries. We then used these common elements to define an international set of standards reflecting those countries demonstrating excellence on the TIMSS population two test. The resulting international standards (benchmarks) were then analyzed in terms of their coherence.
Top achieving countries identified for this
study are named as the ëA-plusí (A+) countries (Valverde
& Schmidt, 2000). The set of A+ countries identified are those
countries that have the highest mean middle school student
achievement (total score) without identifying more than five
additional TIMSS countries that could be statistically equivalent to
them; that is, to include the next lower achieving country would make
it necessary to include more than five additional countries that have
achievement statistically equivalent to theirs. In mathematics, six
countries were identified, Singapore, Korea, Japan, Hong Kong,
Belgium (Flemish), and the Czech Republic, and in science, four
countries, Singapore, the Czech Republic, Japan, and Korea.
The data used to develop the international
benchmarks come from the curriculum component of TIMSS. The data
derive from the procedure known as General Topic Trace
Mapping (GTTM) (Schmidt, McKnight, Valverde, Houang &
Wiley, 1997).The respondents to the GTTM were education officials
(typically curriculum persons in the national ministry) of each
nation who, utilizing their national content standards or an
aggregate of regional standards, indicated for each grade level
whether a content topic was intended or not. The result was a map
reflecting the grade level coverage of each topic for each country.
This study only reports on the mathematics and science topic coverage
from first through eighth grades.
The topic trace maps were available for each
of the A+ countries. While none were identical they all bore strong
similarities. The following procedures were followed to develop an
international benchmark.Firstthe mean number of intended topics at
each grade level was determined across the A+ countries. Next the
topics were ordered at each grade level based on the percent of the
A+ countries that included it in their curriculum. Those topics with
the greatest percentage were chosen first and only as many were
chosen as were indicated by the mean number of intended topics at
that grade level.
Several caveats are necessary in order to more carefully characterize the procedures. The first caveat involves the situation 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 that case all those tied topics were listed as optional with an indication of the number of topics to be selected from the optional list. Second, 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. In this way the optional topics can be thought of as those 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.
The data on US content standards in mathematics and science were collected from three sources. First, the mathematics and science standards developed by professional organizations in the US were used. This included the National Science Education Standards released in 1996 by the National Academy of Science and National Research Council, and the Curriculum and Evaluation Standards for School Mathematics prepared by the National Council of Teachers of Mathematics in 1989. Second, a sample of 21 state mathematics and science standards in effect during the 1999-2000 school year was used. Finally, a sample of district standards collected from approximately 50 districts was used. These data indicated topics intended for instruction at each grade level through eighth grade.
Content Coding Methodology
The sampled national, state and district
standards were subjected to a different content analysis procedure
than that used for the development of the international benchmarks.
The TIMSS curriculum analysis developed a formal systematic document
analysis methodology which produced curriculum data on about 50
countries utilizing content standards and textbooks but only at the
grade levels associated with the three TIMSS populations (Schmidt,
McKnight, Valverde, Houang & Wiley, 1997). It was this method of
content analysis that was used to generate the US data using state
and district content standards.
Coders (graduate students with degrees in
mathematics, engineering, and the various sciences) were provided
with days of training and pages of documentation to master the
techniques. The method described elsewhere (Schmidt, McKnight,
Valverde,Houang & Wiley, 1997) includes dividing the content
standards into small segments called blocks. After defining
the blocks the actual instructional material in each block is
described using categories from the TIMSS curriculum frameworks,
i.e., coders identified each block's content as to 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.
These trained coders produced a detailed line-by-line, page-by-page content analysis of the standards using the framework content codes. These same content codes 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 rather only their use by experts to make informed judgements.
MATHEMATICS
Figure 1 portrays the set of topics for grades one through eight that represents only those common topics that were intended by a majority of the A+ countries. The data suggest a three-tier pattern of increasing mathematical complexity. The first tier included an emphasis primarily on arithmetic, including whole number concepts and computation, common and decimal fractions, estimation and rounding. It was covered in grades one through five. The third tier consisted primarily of advanced number topics including exponents, roots, radicals, orders of magnitude, and the properties of rational numbers; algebra, including functions, and slope; and geometry including congruence and similarity and three-dimensional geometry. This tier was covered in grades seven and eight. Grades five and six appear to serve as an overlapping transition or middle tier with continuing attention to arithmetic topics (especially fractions, decimals, estimation, and rounding) but with an introduction to the topics of percentages; negative numbers, integers and their properties; proportional concepts and problems; two-dimensional coordinate geometry; and geometric transformations, all of which except for percentages were also topics found in the third stage.Thus, grades five and six serve as a point of transition where attention to topics such as proportionality and coordinate geometry led to the formal treatment of algebra and geometry, characteristic of the third stage.The implied curriculum structure also included a small number of topics (six) that provided a form of continuity across all three stages. These topics (measurement units; perimeter, area, and volume; algebraic equations including the representation of numerical situations and the informal solution of simple equations; data representation and analysis; and basic two-dimensional geometry including points, lines, angles, polygons and circles) appear to have insured stability across the three tiers, serving perhaps as ëbuttressesí supporting the overall curriculum structure. Those ëbuttressesí essentially included the fundamentals of algebra, geometry, measurement and data analysis and by the implied breadth of these topics could move from their most elementary aspects to the beginnings of complex mathematics.
The ëupper triangularí appearance of the display in
Figure 1 implies an hierarchical sequencing of the topics in the top
achieving countries over the first eight grades. As discussed in the
preceding paragraphs the sequencing moves from elementary to more
advanced topics in a way that appears to be based mostly on the
inherent logic of the involved mathematics itself. Not only is
the progression of the topics over grades logically consistent with
the nature of the mathematics but that same progression culminates at
seventh and eighth grade in more rigorous topics than what is usually
intended in the US at least for the majority of students. These two
characteristics combine to provide an example of the coherence and
rigor discussed in an earlier section of this paper.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 simply not developmentally ready. On the
other hand, some topics are intended to be introduced early in the US
curriculum because it is supposedly good for motivational purposes or
developmentally lays the groundwork for its own later coverage, even
though this might be inconsistent with the mathematics itself. 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
prerequisite knowledge had to be covered and mastered first and not
primarily because of other issues. Clearly such issues can 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.
Another pattern identified from Figure 1 is the duration over which a topic was covered in the curriculum of the A+ countries ó mathematics topics were intended for an average span of three years. Eight topics out of the 32 were covered for five or more years. These included arithmetic topics such as whole numbers; measurement topics including units as well as perimeter, area and volume; data analysis; equations; and two-dimensional geometry including points, lines, angles, and circles. Five out of the 32 topics were covered for only one year -- measurement, estimation and errors; congruence and similarity; rational numbers and their properties; functions; slope and trigonometry. These five topics were first covered in middle school. These same five topics appear in the mathematics curriculum of upper secondary school although Figure 1 does not show this. Topics such as rounding, properties of operations of whole numbers, estimating quantity and size, and properties of common and decimal fractions were among the 11 topics that were in the A+ curriculum for two years. They were generally introduced and finished in two consecutive grade levels (e.g., introduced at grade four and finished at grade five). As previously mentioned, there was only one topic ó ëmeasurement unitsí ó that was found to be intended by A+ countries from grade one through seven. It was the longest lasting mathematics topic.
The topics displayed in Figure 1 are those covered by the majority of the top achieving countries ó at least four out of the six. The potentially misleading aspect of these data is that they do not represent a complete curriculum for any one of the six top achieving countries even in a composite sense. There are too few topics represented at each grade level. To represent a complete mathematics curriculum (at least in the composite sense) an average of about three topics would have to be added at each grade level in addition to those listed in Figure 1. The actual range of additional topics as noted in Figure 1 goes from one topic (in grades four and five) to as many as six (in grades two and seven).
Unfortunately there is an ambiguity that arises in attempting to implement such a strategy, because for the most part the choice of additional topics is not clear. After including the topics covered by the majority of countries, the countries split themselves fairly evenly over a larger-than-needed set of topics. Very often the split among topics accounts for 50 percent or fewer of the countries. Choosing additional topics to round out a composite curriculum would be completely arbitrary.
To accommodate this ambiguity, but without creating false impressions
regarding the composite curriculum, table 1 lists those topics which
were covered by the largest number of A+ countries even though this
is less than the majority of those countries. Using the number of
additional topics needed at each grade level from Figure 1 together
with the list in Table 1 provides a better sense of how the composite
curriculum would actually look.The important issue for the coherence
argument is to determine if the addition of these topics to Figure 1
alters in any major way the original pattern. We took the most
conservative approach to answer this question and put all of the
potential additional topics listed for each grade level in table 1
into figure 1. To do this, of course, alters the basic conception
upon which figure 1 was based ó 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 type of problem ó it suggests more topics
than would be typical for the A+ countries. However, the inherent
ambiguity permits few if any reasonable alternatives.
The results of this approach are presented in Figure 2. One
interesting and important conclusion is that only two topics found in
table 1 were not a part of the original 32 topics represented in both
Figures 1 and 2. They are real numbers and validation and
justification both of which were intended for coverage only at grade
eight and only by three of the A+ countries.
The basic three tier structure is once again visible in Figure 2. The essential coherence is preserved. A major difference is that the duration of the coverage is longer for several topics. For example, the six ëcurricular buttressesí now are intended across an even larger number of grades, mostly by being included in earlier grades. Actually, the biggest change is in the increase in the number of ëbuttressesí as a result of the inclusion of two estimation topics and the topic properties of whole number operations. The only other significant change is that three-dimensional geometry becomes part of the overlapping transitional tier two.
To compare with the composite curriculum for
the A+ countries, Figure 3 illustrates the intended coverage for the
original NCTM standards (NCTM Standards, 1989). The mathematics
topics listed are the same and they are displayed in the same order
as they were in Figure 1 for the A+ countries. The pattern for the
NCTM standards is totally different from that of the top-achieving
countries. For example, the upper triangular structure of figure 1
disappears.
A caveat important in interpreting Figure 3 is the design of the NCTM
standards. It has a ëclusterí organization ó
content standards are distributed in clusters of grades including
kindergarten through grade four, grades five through eight, and
grades nine through twelve. This structure results in blocks of
topics being assigned to each cluster ó in particular, to
grades one through four and grades five through eight for the grade
range we are considering. This grade-level ambiguity inherent in the
standards had to be resolved in some fashion. The NCTM Standards do
not contribute to the resolution of this issue since according to
their own statement (NCTM Standards, 1989, p. 252), 'The standards is
a framework for curriculum development. However, it contains neither
a scope-and-sequence chart nor a listing of topics by specific grade
level.'
Based on anecdotal data on how these standards have been used by
states and districts in developing their standards and given the
content of US textbooks we have taken the standards at face value.
For example (NCTM Standards, 1989, p. 48), when they state ëIn
grades K-4, the mathematics curriculum should include
two-and-three-dimensional geometry so that students can,í we
have made the assumption that first grade teachers will assume
reading this that they are to cover these topics and so will second,
third, and fourth grade teachers. We assume that only if told
otherwise will teachers at each of these grade levels believe they
are not to cover this and other such specified topics. The TIMSS data
on teacher coverage of topics supports this conjecture (Schmidt,
et al, 1999).
There are noticeable blank areas for grades one through four ó
for the topics of properties of common and decimal fractions,
percentages, proportionality concepts, proportionality problems,
negative numbers, integers, and their properties, exponents, roots
and radicals, rational numbers and their properties, and slope and
trigonometry. This implies that these topics are not listed as
content topics required of US elementary students. For the other 24
out of 32 mathematics topics, the NCTM standards intend coverage of
them across all of the first eight grades, at least according to our
conservative interpretation.
Specific information of how long a topic will remain in the curriculum is ambiguous as a consequence of the clustering of grades,. However, taking the standards at face value would suggest a range of from four to eight grades with the preponderance of topics being included in all eight grades.
The 32 topics which define the body of Figure 3 include all of the topics intended for coverage by a majority of the top-achieving countries. The TIMSS mathematics framework contains a total of 44 topics. Of the remaining 12 topics not included in Figure 3, only two are needed to describe the A+ composite curriculum. For the NCTM standards an additional seven of the 12 remaining topics in the framework are intended for coverage including the ambiguous ëother.í Over grades one through eight the NCTM Standards intend the coverage of 39 of the 44 TIMSS mathematics topics with slightly over half of them intended across all eight grades.
Figure 4 illustrates the results of applying
similar statistical techniques to the mathematics content standards
of 21 states.The criterion for topic inclusion at each grade level
for the A+ countries was a simple majority of those countries.
However, with only six countries that criterion effectively was a
two-thirds majority. A similar two-thirds majority was applied in the
case of the 21 states. Several of the states specified their
standards according to the cluster model described for NCTM but
others had grade-specific content standards.
The resulting pattern more closely resembles that of the NCTM
standards than it does that of the A+ countries. This is not
surprising since the NCTM standards were used as a model in the
development of many state standards (Blank, et al, 1997). The
state standards do not reflect the three-tier structure described
previously. The majority of the 32 mathematics topics identified for
the grade one through eight students in the A+ countries are likely
to be taught to American students repeatedly throughout the eight
years of elementary and middle school. In fact, the average duration
of a topic in state standards was almost six years. This is twice as
long as for the A+ countries.
The result of the larger average duration is readily visible in
Figure 4. The silhouette of Figure 1 has been superimposed on Figure
4 to outline the A+ pattern and make the comparison more obvious. The
state standards generally increase the duration of a typical topic by
intending coverage at earlier grades -- even for the more demanding
topics such as geometric transformations, measurement error,
three-dimensional geometry and functions. These topics were all
ëintendedí for coverage as early as first grade by a
two-thirds majority of the examined state standards. These same
topics were first intended for coverage in middle school for the top
achieving countries -- that is, for grades six, seven and eight.
Clearly these results raise the suggestion of a lack of coherence
even if the claim is that these topics are only presented initially
in an introductory, elementary fashion.
The intended coverage of the properties of operations of whole
numbers (such as the commutative and distributive properties) at
first grade, the same time they are beginning to study the basic
operations, is another illustration reflecting a lack of coherence in
the state standards. This topic is first introduced at grade four in
the top achieving countries (certainly, in the interest of a fairer
comparison, no earlier than grade three).
No topic was intended for coverage at all eight grades for a majority of the top achieving countries yet 11 topics were intended for such a broad coverage for at least two-thirds of the 21 states. All of this implies an arbitrary, laundry-list approach to be mostly characteristic of the aggregate state standards. This hypothesis is further supported by the fact that two-thirds of the 21 states do not agree as to the placement of three fundamental topics ó rounding and significant figures; the properties of common and decimal fractions; and slope. For slope it is not even intended for coverage until high school in the majority of the 21 states. By international standards this at least suggests a lack of rigor. For the other two topics the data reflect not that they are intended for coverage at a later grade but rather reflect a lack of agreement on where to place these topics in the sequence of the first eight grades.
Figure 5 displays the sequence of intended coverage for each of the 21 states' mathematics standards separately. The order of the topics is the same as in Figure 1. This is done to show that the results of Figure 4 using a composite of the 21 states does not appreciably distort the reality of individual state standards. 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. This is somewhat of an over-generalization but the patterns do suggest that something like such a general conclusion would not be unreasonable.
Figure 6 is the composite of approximately
50 districts' mathematics content standards constructed in a manner
similar to that for the A+ countries and the 21 states. These
districts were chosen randomly from one of the 21 states (State E in
Figure 5). This state had policies in place to reinforce and assure
coverage of the state standards. There is, however, not much
similarity as one compares this pattern with State Eís
mathematics standards in Figure 5 especially considering the fact for
only for one topic and only at two grades do all districts conform to
the state standards. This indicates a general lack of alignment
between the state and many district standards.
The district data, as do the state data, reflect a lack of
coherence in the disciplinary sense discussed earlier. Several of the
same ten topics intended for coverage by two-thirds of the states at
all eight grades are similarly intended for such coverage in the
majority of the districts in State E; still others are intended for
coverage in seven of the eight grades. There are other manifestations
of the lack of coherence for the composite district standards in
state E. Coverage of whole number operations is intended for even two
more grades than was the case for the states. Functions and
three-dimensional geometry were intended to be introduced at first
grade; this was similar to a majority of the states but in
contradiction to the majority of A+ countries (where they were first
introduced in grades seven and eight).
A+ countries have a somewhat different
tiered structure for science than was the case for mathematics. We
present this example primarily as an illustration that the coherence
argument is not idiosyncratic to mathematics. In mathematics the
three tiers corresponded to progressively more complex topics that
built on those in the previous tier and reflected the somewhat
hierarchical nature of mathematics. Only a few topics received
continued attention across all three tiers. Science as a discipline
also has somewhat of a hierarchical structure but not as strictly as
for mathematics. This is especially true for school science which is
comprised of four disciplines ó physics, chemistry, biology
and earth science.
Science is a combination of topics from these four different
disciplines in elementary school, and even for middle school, in most
countries. At the most theoretical level there are inter relational
implications for knowledge across the disciplines. For example, the
chemical basis (chemistry) of cellular activity (biology) includes
the structural fact that living cells are comprised of quarks and
electrons (physics) and that cells require certain minerals (geology)
for their metabolic functioning (bio-chemistry). However, this does
not imply a strict hierarchical ordering of topic coverage --
especially when intended school coverage focuses more on the
descriptive aspects of science.
One implication of this is that one might not expect a priori
to have as hierarchical an ordering of science topics as for
mathematics at least at the elementary and middle school levels.
Another possibility is that topics might remain in the curriculum
over an extended period of time as the intended coverage progressed
from more elementary descriptive aspects to more theoretical and
explanatory aspects; the progression of topics over grades then would
reflect the more theoretical inherent logical structure.
The first of these two possibilities would imply more of an ad hoc structure across all of the science topics, one that might be formally structured within any one of the four disciplines but appear arbitrary when they were combined. The second possibility would likely manifest itself in an upper-triangular structure similar to mathematics, but one in which topics remained in the curriculum relatively longer reflecting not redundancy in intended coverage but a within-topic progression from more descriptive to more theoretical.
The data from the top achieving countries are consistent with this
second possibility. The pattern of Figure 7 suggests that for science
the key question is at what grade a topic enters the curriculum; once
intended it essentially remains intended for all subsequent grades.
The structure underlying Figure 7 reflects a "spiral" approach but
one that is ëstaggeredí over topics ó that is, not
all topics are spiraled from the same starting point.
There is, somewhat surprisingly, no agreed-upon science topic to be
introduced in grades one and two among the A+ countries. Three of the
four A+ countries do not have formal science as a part of their
curriculum until grade 3. As a result of the spiral approach seen in
Figure 7, most of the science topics introduced in the primary grades
(grades three and four) continue in the curriculum through grade
eight. These topics appear to serve the same role as the
ëbuttressesí did in mathematics. They represent some of
the most fundamental concepts of science: (1) the classification of
living organisms and their systems -- plants, fungi, animals, organs,
tissues, and life cycles; (2) the classification of earthís
physical features -- rocks, soil, and bodies of water; (3) the
classification of matter as well as physical properties and changes
of matter; and (4) different forms of energy -- light, electricity
and heat. These fundamental topics come from physics, biology and
earth science and run through the first eight years of schooling
(although for three of the A+ countries this coverage is not intended
to begin until third grade).
All of these are broad topics for which the intended coverage could
deepen from year to year. This is unlike a topic such as whole number
operations for which after a point there would be no deepening only
repetition. Those operations can be placed in a more theoretical
context such as the real number system. However, in the mathematics
framework the latter is captured by a different code and would as
such represent a change of topics. There are similar instances in
science of deepening the coverage of a topic by additional
theoretically linked topics, but there is also considerable deepening
of coverage that can occur within a single topic, probably more so
than in mathematics. However, it is clear for these 12 basic topics
that they are deemed important and fundamental enough among the top
achieving countries to be included at the start of the science
curriculum and to continue in some fashion through grade eight.
The lower middle grades (grades five and six) continue these same topics (with the exception of light and electricity) but introduce additional and more complex topics from each of the three sciences involved. Biology concepts presented in the primary grades dealing with the classification of living organisms and their morphology provide a foundation for the study of (1) within-organism development and (2) the interaction of living organisms both with other organisms and with their environment. These latter topics mainly deal with what has been termed ecology and environmental science. Within-organism development focuses on the basic aspects of life cycles themselves and on reproduction.
Some earth science topics are also related to concurrent life science topics for the lower middle grades. While students are expected to learn how living organisms interact with their environment; supportive earth science topics for these grades are weather and climate and the composition of the earth. The solar system is the other major thrust of the earth science curriculum in grades five and six. In the physical sciences, building on the study of matter and energy begun in grades three and four, the top achieving countries intend their students to study magnetism, types of forces (such as gravity), and time space and motion at grades five and six. The latter two topics are conceptually related to the solar system topic included in these lower middle grades.
In the upper middle grades (seven and eight) the top achieving countries' science curriculum intends students to study chemistry and related topics for the first time. This includes atoms, ions and molecules; explanations for physical changes (including boiling, freezing and dissolving); chemical changes such as oxidation-reduction; and explanations for such chemical changes (e.g., covalent bonding and electron configurations). Physics topics also are included ó sound and vibration; types and sources of energy such as potential and kinetic; and the dynamics of motion. Lightí also returns as a topic, having been introduced in grade three but absent after that.
Earth science topics introduced in grades seven and eight build on the concomitant physics and chemistry topics introduced in those upper middle grades. These include physical cycles such as the rock and water cycles; plate tectonics; the atmosphere; and conservation topics such as pollution. In the life sciences, using newly introduced physics and chemistry topics and building on the study of organs and tissues, the A+ countries introduce biochemistry and the physiology of organisms -- covering such topics as sensing and responding, cells (including cell membranes, mitrocondra and vacuoles), and human nutrition.
In one sense the science curriculum is similar to mathematics in having an upper triangular structure with three tiers. However, science's first tier serves as the unifying element (the curricular ëbuttressí) since those fundamental topics introduced in grades three or four continue throughout the rest of the first eight years of schooling. The lower middle grades (five and six) serve as the second tier and focus mainly on ecology and environmental science (supported with topics from both biology and earth science), the solar system (supported by physics topics), and magnetism. The upper middle grades (seven and eight) provide the third tier as they did for mathematics; they are focused in the top achieving countries on chemistry, physics, biochemistry, physiology and earth science topics that build on chemistry.
The average number of grades for which a topic is intended is three ó the same as for mathematics. This is true even though the organization of the science curriculum for the A+ countries appears to be based more on a spiral model than was true for mathematics. Six was the maximum number of years a topic was in the curriculum since no formal science instruction was intended at grades one and two.
Another way to examine the data in Figure 7 is to organize the
same 41 topics into the three major disciplines (treating chemistry
and physics as one). This is possible because school science is
comprised of topics from four distinct disciplines. This
reorganization is shown in Figure 8. The biology data show a
progression from taxonomy and morphology to ecology and environmental
science to physiology and biochemistry; they thus follow the three
tier structure. The earth science progression similarly moves from
description and taxonomy to the various physical processes that
govern the solar system and, in particular, the planet
earth.
The physics and chemistry topics progress from descriptive
classification and the properties of matter to physical changes of
matter; and from energy and forces (such as gravity) to basic
chemistry, other forms of energy and motion. The earlier discussion
around Figure 7 shows how each of these separately coherent
within-discipline topic sequences are integrated across the four
disciplines.
We now turn to the aggregate A+ or composite curriculum for science.
As before with mathematics there is an ambiguity which needs
resolution. Table 2 lists potential additional topics that can be
used to make up the requisite number of topics. In science all these
potential additional topics were covered by two of the four top
achieving countries.
Figure 9 corresponds to Figure 2 for mathematics. Science had an
appreciably lower ratio of the number of additional topics needed to
define the composite curriculum for the A+ countries to the number of
such topics potentially available than did mathematics. This, of
course, is owed at least in part to the smaller number of A+
countries for science. As a result the pattern in Figure 9 is very
distorting. For example, only two of the 19 additional topics listed
for grade six are needed. Nevertheless, to be consistent the large
numbers of optional topics are all included.
The general pattern of Figure 7 still holds with four possible
exceptions. Magnetism, atmospheres, sound and vibration, and sensing
and responding now become part of the fundamental set of topics that
are buttress-like ó essentially being in the intended
curriculum for all six grades. Magnetism and sound and vibration join
other energy topics such as light, electricity, heat and temperature
while sensing and responding (the nervous system and brain) join
reproduction as part of the early life systems to be
covered.Atmospheres joins rocks, soil and bodies of water as part of
the descriptive features of the earth.
The only other change similar to mathematics is that numerous topics
move down a grade or two in intended coverage but still preserve the
upper triangular structure. As a final comment here it is worth
noting that the physics topic of energy types, sources and conversion
stays pretty much in parallel here (as in Figure 7) with the biology
topic of energy handling which deals with energy, its storage and
transformations including photosynthesis. The logical connection is
obvious even though the two topics come from different disciplines.
This is still another example of the kind of coherence for which we
have been arguing.
In contrast to the upper triangular pattern
of the A+ countries, Figure 10 reveals an entirely different pattern,
or perhaps more accurately the absence of a clear pattern, for
science topics in the US national science education standards. We
observe a more comprehensive curriculum in the US in contrast to the
A+ countries just as was the case for mathematics. Science was
clearly a higher priority in grades one and two in the US than in the
top achieving countries. After grade two the US national standards
suggest including almost 30 of the 41 topics; the majority of A+
countries only intended the coverage of 12 in grade three.
Figure 11 shows the composite science curriculum for the same 21 states in Figure 4. There are four topics that were intended to be taught from grades one through eight by at least 83 percent of the states. Compared to the A+ countries, there are two overlapping topics with a similar range of intended (coverage recognizing that for those countries science did not begin until grade three) ó physical properties and plants and fungi.. Two other topics had a duration of eight years -- earth in the solar system; and energy types, sources and conversions. The energy topic is particularly noteworthy since it is in the A+ composite curriculum only at grade eight and even for the more conservative approach presented in Figure 9 it is first covered in grade six. This is in marked contrast to coverage in grade one for the majority of states; given the complexity of the topic even at its most elementary level (the definition of energy), this suggests a rather ad hoc placement.
The silhouette of the A+ intended topics coverage is
superimposed on the data in Figure 11. The contrast this presents
reveals patterns such as those just described for the topic of energy
types. In general, the duration of the intended coverage is longer
for most topics in the states compared to the A+ countries; the
difference results from the earlier introduction of these topics.
Figure 12 shows the patterns for each of the individual 21 states.
These patterns are mostly similar to the national standards but 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),
with only three exceptions all of these states have many topics
introduced at grade one that last through grade eight. The most
obvious example is the standards for state J; 41 topics are intended
for coverage starting in grade one. Based on these individual
patterns it seems clear that the ëspiralí approach
evident at least in part in the composite A+ curriculum dominates in
the state standards; it seems, in fact, to be the primary organizing
principle for almost all topics. However, unlike for the A+
countries, the spiral in these states is anchored at grade one for
almost all topics. There is no staggering of the spiral's starting
point across topics as there was for the A+ countries.
The National Science Teaching Association's scope,
sequence and coordination document has long argued for such a spiral
approach to education (Aldridge, B., 1992). However, we suggest that
without a ëmeta-organizationí of the spiral principle
that recognizes the inherent logical structure across the topics
ó both within and across the four disciplines ó the
pattern seen in Figure 12 results. In such a case the
ëspiralí seems to collapse into a circle since it does
not seriously progress and sends those who follow it repeatedly
around a set of topics. Early introduction of topics may appear to be
a good way to ëjump-startí the learning process, however,
because of limited prerequisite knowledge, and lack of experience and
an underdeveloped approach to scientific thinking, the result may
well be the development of misconceptions which later can impede
further learning.
We have argued here that coherence is one of the most critical -- if not the single most important -- defining elements of high quality standards. We defined content standards' coherence in terms of the discipline that underlies the school subject matter articulated in those standards. We suggest that standards are coherent if the standards specify topics, including the depth at which the topic is to be studied as well as the sequencing of the topics both within each grade and across the grades, in a way that reflects an inherent logic of the underlying discipline.
We do not argue that this characteristic demands that there is only one set of coherent content standards for a particular discipline such as mathematics. However, we do believe that there would likely be a limited set of such standards. It is our belief that such coherence in standards is critical to learning for understanding. In Brunerís terms understanding in fact implies at least at some level that the structure of the discipline has become visible to the learner so she or he can move beyond its particulars. We suggest that one way to facilitate such learning is by making the inherent logical structure of the discipline more visible both to teachers and students.
Content standards are policy instruments used by countries around the world to articulate the vision of a subject matter discipline to its educational system; this includes not only teachers and students but parents and the general public as well. We suggest that one of the more important policy questions for the US in evaluating state and district content standards is whether those standards reflect such a coherence. We suggest that if they do not the result will likely be more of a rote memorization of particulars without the deeper understanding necessary to generalize beyond those particulars rather than learning for understanding and generalization. We believe this is in general true of US mathematics and science standards. The data presented in this paper on US standards ó professional, state and district alike ó reflect a 'laundry list' or ad hoc model of topic organization rather than reflecting a discipline-based logical structure. We believe that this is not unrelated to the fact that in TIMSS the US did comparatively poorly internationally especially in the later grades ó grade eight and the end of secondary school.
In the 40-some TIMSS countries the middle grades were where there was a shift from elementary mathematics and science to the more formal aspects of these disciplines ó where, in fact, the inherent structure became more cogent. These are exactly the grade levels at which the US fell behind the other countries. Earlier TIMSS performance (fourth grade) was not as bad in cross-national comparisons. Is this coincidence or are the two related? Questions of causality are impossible to answer with survey data but a series of formal statistical analyses lend support to the argument that this is not coincidental (Schmidt et al., 2001).
Data for the countries that clearly performed the best in TIMSS were used to examine a composite curriculum for those countries. Organizational structures for the mathematics and science topics in this composite curriculum were inferred from the data for the majority of the top achieving countries. These organizational structures seemed to be based on the principle of coherence we have elucidated here. The resulting upper triangular structure evident for both mathematics and science suggests sequencing over grades that does not treat the topics as interchangeable parts placed arbitrarily on a grade level grid; instead the topics appear to be sequenced to reflect hierarchical and logical structures of the disciplines. Again, we are not arguing that either is the only structure that would imply coherence for their respective discipline or even that either one is the best structure. Instead we are simply suggesting each is one of a relatively small number of such coherent structures for mathematics and for science.
The contrast is striking when one examines the national professional standards developed by NCTM for mathematics and by the NRC for science. The contrast is equally striking when one examines the composite standards for 21states, for more than 50 local districts within one state, or the individual standards in each of the 21 states. Not only is the organizing principle underlying the structure of the US unlike that of the A+ countries but the organizing principle for the US standards seems to be qualitatively different rather than simply differing in degree.
Even the sheer number of topics intended for coverage in the US is visually overwhelming as reflected in the displays presented here. This is the 'mile wide/inch deep curriculum' as we have called it elsewhere (Schmidt, McKnight and Raizen, 1997).Further, the organizing principle (if one can call it that) seems to be 'include almost every topic at almost every grade.' If the US standards differed by degree one could argue about alternative sequences that might contribute to coherence. However, the differences are so stark and the US structure so diffuse and seemingly arbitrary that such an argument seems inappropriate.
The word 'diffuse' above was chosen carefully and is suggestive of a scientific analogy. Diffusion is defined in science as the process whereby particles intermingle as a result of their spontaneous movement caused by thermal agitation. Substituting 'topics' for 'particles' and whatever word you believe describes the process by which standards are developed in the US for 'thermal agitation' -- political compromise, shared decision making, etc. -- while remembering that the word 'spontaneous' means to proceed without external constraint or on a momentary impulse, we suggest that 'diffuse' is a good adjective for US content standards, at least in mathematics and science. Certainly the result is something like the opposite of 'coherent.'
Further empirical support that coherence in content standards results in greater learning and deeper understanding must await additional study. TIMSS' analyses show strong relationships cross-nationally of content standards to both what teachers teach and what students learn (Schmidt et al., 2001). They also demonstrate how different US standards and classroom instruction are in the aggregate from other countries, both in terms of the sheer number of topics and in terms of how rigorous or demanding those topics are for students (Schmidt, et al, 1999). This all suggests that our hypothesis on the importance of coherence as we have defined it is promising given the existing data.
US policymakers must decide whether to await more definitive proof or to act on the evidence we have. One factor on the side of more immediate action is that coherence seems a more logical principle from a subject matter perspective. One implication of this is that discipline specialists (university professors, mathematicians, etc.) should play an increased role in the development of content standards.
Finally, we turn to the empirically least-well supported part of our argument ó that coherence and rigor might only be possible in the US if the standards are national (not federal) in scope. The support for this argument is a combination of logic and anecdotal evidence. Logically, it is easier to produce coherent standards when there is one institutional center responsible for the task. Anecdotally, standards are national for most countries in the TIMSS world and they are regional even in those few countries in which they are not national. We suggest that the reluctance of the US policy community to address this only contributes to the diffuse nature of US standards. This is especially the case given the prominent role textbooks play in this country. Since US textbooks are designed and written for a national market and since they are often the only common element across classrooms they become de facto national standards (Schmidt, McKnight & Raizen, 1997) and, as such, have enormous influence (Schmidt et al., 2001) on what teachers teach. Consequently they have both a direct and an indirect (through teachers) effect on what is contained in US state and district standards. A recent study (Valverde, Bianchi, Wolfe, Schmidt, & Houang, in press) provides data on the diffuse nature of these books.
This
diffuse nature of textbooks perpetuates itself in one grand
ëcatch-22í in a system without national standards to
mitigate it. Certainly nothing is done to mitigate the diffusion of
curriculum content standards in mathematics and science. More
importantly, nothing is likely to be done until US policy makers
reconsider the importance of subject matter coherence relative to the
inertia of current practice and ideologies of child development
primacy. Curricular coherence seems essential and seems attainable if
it becomes a common understanding, an active goal, and the source of
necessary changes (for example towards national efforts to produce
coherence to guide schooling).
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