Georgia Performance Standards in Mathematics

Response by Bradford R. Findell

Department of Mathematics Education

University of Georgia

March 8, 2004

with input from Paola Sztajn, Denise Mewborn, Jeremy Kilpatrick, Andrew Izsák, and Andy Norton, Department of Mathematics Education; and Elliot Gootman, Ted Shifrin, John Gosselin, Shuzhou Wang, Calvin Burgoyne, Robert Rumely, Tawanda Gwena, Dhandapani Kannan, Sybilla Beckman, Paul Wenston, and Jason Cantarella, Department of Mathematics.

Introduction

This response to the Georgia Performance Standards (GPS) in mathematics was prepared at the request of Associate Dean Karen Watkins of the College of Education and Associate Dean Charles Kutal of the Franklin College of Arts and Sciences, both at the University of Georgia. It is based on discussions with, input from, and feedback from mathematicians, statisticians, mathematics education researchers, graduate students in mathematics and mathematics education, and school mathematics teachers. Not all of the points in this response were discussed by all groups, but participants were unanimous on many points, and groups and individuals offered additional points of their own, some of which are included here. The general sense from all participants was the following:

The QCCs needed changing in the proposed directions,

The current draft needs monumental work, and

The implementation plan is severely lacking.

We wish to emphasize that without substantial revisions of the current draft, without rethinking the timeline for implementation, and without substantial resources for professional development, this revision of Georgia's mathematics standards could be a disaster for Georgia's teachers and students. Furthermore, poorly written standards and tests may make Georgia a target for national ridicule, as has happened in other states.

We suggest that, at the very least, the state adopt a more reasonable timeline and involve mathematicians, statisticians, and mathematics education researchers as writers at every grade band. We know that the Regents' Advisory Committee on Mathematical Studies has offered to help. We suspect that many faculty, departments, and committees throughout the University System of Georgia would also offer help, particularly with the recently funded project, Partnerships for Reform in Science and Mathematics, in which both the Department of Education and the Board of Regents are key partners.

Summary

We are in complete agreement that the Quality Core Curriculum in mathematics needed to be changed-even completely reconceived. There were too many objectives, and they focused mostly on skills rather than concepts, reasoning, and connections. There was too much repetition from grade to grade and not enough depth at any grade. In short, they were "a mile wide and an inch deep."

The principles behind the new Georgia Performance Standards are appropriate and well founded: balancing concepts and skills; emphasizing the process standards of the National Council of Teachers of Mathematics (NCTM): reasoning, problem solving, communication, connections, and representation; presenting content standards along with tasks, student work, and teacher commentary; and having high expectations for all students. These are ambitious standards. They are intended to be "world class," and with substantial work it may be possible to achieve this goal.

The current draft, however, does not deliver on these promises and does not reflect these principles. In addition to the documents themselves, we are concerned about the processes for feedback, revision, and implementation. In particular, we see the following flaws:

The GPS documents do not give sufficient sense of what is intended under each standard.

The standards and tasks contain numerous errors of mathematical language and content.

The NCTM process standards should be integrated throughout the content standards, but they appear to be separate and unrelated.

The documents appear superficially to fit the "Japanese model" but not enough work has been done to understand that model and adapt it for use with Georgia's teachers and students. For example, many of the standards are miscategorized because the Japanese content categories did not fit well with the NCTM content strands.

At middle school and high school, because of significant differences in sequencing, there are no textbooks available to support these standards. Available translations of Japanese texts do not include materials for teachers. The suggestion that middle grades teachers will merely supplement with geometry lessons misunderstands how mathematics textbooks are used in this country and how difficult it is to learn to use new materials. At the high school level, because of the highly-specified sequencing of topics, currently-available integrated texts will not align with the draft GPS. Furthermore, there is not sufficient time for commercial publishers to develop quality textbooks that align well.

In writing both standards and sample tasks, the writers appear to have made little use of recent curriculum and assessment development projects in the U.S. We point, in particular, to the NCTM Illuminations project, the New Standards and Balanced Assessment projects, and the 12 curriculum projects funded by the National Science Foundation (NSF) during the 1990s to support NCTM's 1989 standards. For information about these projects, see http://www2.edc.org/mcc/.

The standards do not do justice to data analysis and, in fact, appear to be a step backward from the QCCs in that regard. Data analysis appears to be ignored at K-5. At high school, most of the content listed under the data analysis strand is probability or algebra. Not until Math IV are students expected to reason with data, and there are no opportunities for using data analysis to make sense of, say, algebra.

The process for developing the GPS ignores what has been learned in more than 15 years of standards development throughout the country. In particular, the best processes have involved, in both writing and oversight, not only teachers but also mathematicians, statisticians, mathematics education researchers, curriculum developers, and mathematics supervisors. In the recent and not-so-recent history of mathematics education, there are many initiatives that have failed by ignoring one or more of these groups.

The implementation plan does not give sufficient time for teacher learning or materials development. In particular, a two-day workshop for one teacher from each school is so small and brief an intervention, it will be meaningless. We note that Japan adopted a more gradual implementation plan in 1990, for changes that were minor. The proposed GPS, in contrast, represents a radical (and untested) departure from current practices in Georgia's schools.

There appears to be an insufficient process for sorting, synthesizing, and responding to feedback on the various draft standards that are or will be available over the next several months. Overlapping the revision and comment periods merely increases these difficulties and wastes resources.

We elaborate these comments in the sections that follow. These constructive criticisms are offered in hopes of improving both the documents and the process. In most of these sections, we offer both a description of the problem and a suggestion for the solution. We also provide specific examples of how the standards, the "elements," and the tasks might be improved.

Quality

The draft documents do not communicate well. They are incomplete, oftentimes they do not make sense, and they need serious editing. There are many errors of mathematical language, caused perhaps by poor translations of the Japanese standards and texts. Many standards are included under inappropriate content strands. Few of the standards are sufficiently elaborated or well illustrated by tasks. Furthermore, while the process standards should be threaded throughout the content standards, they are too often missing from the current drafts. In a similar vein, some of the high school courses insist on manipulatives, technology, and multiple representations, but there is no evidence on the pages that follow that this is necessary or even valued.

Our major concern about the documents right now is that the language of the standards is not sufficiently clear, precise, and elaborated to communicate to teachers and to assessment developers what is intended. On first reading, many of the standards made little sense to us, and we could not see how the "elements" under each standard were different from QCC objectives. In the current drafts, it sometimes appears that the QCCs were merely organized by topic so that 57 QCCs could look like only 12 standards. (We know this was not the intent; we are merely describing how the draft might be interpreted.)

As we have worked to understand the documents, we have found that most of the standards, at least through grade 10, are taken almost directly from a not-entirely-accurate translation of the 1990 Japanese standards. Furthermore, the GPS are quite similar to the tables of contents from translations of Japanese texts from 1984, but the draft standards do nothing to demonstrate the richness of the texts. These translations have proven very helpful to us in illustrating what the standards could mean. But how many teachers, assessment developers, and curriculum developers will (or should) go to that sort of trouble?

What is needed

Carolyn Baldree has noted in presentations that these Standards represent a paradigm shift. If this is the case, and if there is any hope that teachers will be able to use these standards to reorient their instruction, the writers of the standards have an obligation to elaborate the standards sufficiently so that teachers can see what it means for a student to understand each topic. The GPS must describe what concepts and skills students are to have learned and demonstrate that reasoning, representations, problem solving, communication, and connections are valued aspects of mathematical activity and learning by threading these process standards throughout the content.

Furthermore, the description of problem solving strand must make clear that problem solving is both a way to apply mathematical knowledge that students already have and also a way of developing new mathematical knowledge. In other words, students do not need to be told ahead of time how to solve every problem. Giving students opportunities to solve problems that they have not seen before is a way to motivate and establish the relevance new mathematical ideas, to engage students in an approximation of authentic mathematical activity, and to learn new content in ways that they are more likely to remember. These principles must be stated explicitly and threaded throughout the standards.

Finally, a committee or group needs to look at articulation across the grade bands, paying attention to the growth in both content and process strands.

Japanese Model

We like the idea of beginning with the "Japanese model," but it is important to recognize that the GPS writers were relying on translations. Furthermore, the U.S. mathematics education community does not have the same understanding as the Japanese community about what it might mean to understand particular topics and what curriculum and instruction might look like to support such understanding.

To cite one blatant example, the content categories do not match the NCTM content strands, and this fact appears to have caused some miscategorization of standards:

Japanese Content Categories	NCTM Strands
Numbers and Calculations (Grades 1-6)	Number
Numbers and Algebraic Expressions (Grades 7-9)	Number and Algebra
Quantities and Measurements (Grades 1-6)	Measurement
Geometrical Figures (Grades 1-9)	Geometry
Quantitative Relations (Grades 3-9)	Algebra, Data, Probability and Statistics

The above table illustrates why some algebra strands were categorized as number and others were categorized as data. Such errors indicate that the writers have not yet made sufficient sense of the Japanese standards and categories.

The current draft GPS reflects the Japanese model only superficially. The writers have not yet done the work to make the standards meaningful, and there have been many mistakes of mathematical content and language. Furthermore, if we really believe in the Japanese model, then the writers should be aware of other significant difference between the U.S. and Japan in educational policies, resources, and practices:

In Japan, there is no tracking. All students take the same courses in heterogeneous classes through grade 10.

Mathematics is optional after grade 10.

In grades 10-12, the 1990 Japanese standards describe both the core mathematics sequence (Math I, II, III, the last of which is calculus) and three additional elective courses (Math A, B, and C).

The Japanese Ministry of Education publishes an elaboration document for each level of school, detailing and illustrating what the standards mean by providing additional specification, a few illustrative examples, discussion of how the goals are related across grade levels, discussion of instructional planning, and elaboration of changes from the previous standards.

New Japanese standards are adopted with a sensible and gradual implementation process. For example, the "1990 Japanese mathematics standards" were released by the ministry of education in 1989 and implemented as follows:

Kindergarten 1990

Grades 1-6: 1992

Grades 7-9: 1993

Grades 10-12: 1994

This implementation was for changes that appear to have been relatively minor, at least through grade 10. (Evidence: Translations of 1984 texts fit the 1990 standards almost perfectly in grades 7-9 and somewhat well in grade 10.)

In 2000, in response to concerns that there was too much pressure on the students, the Japanese mathematics standards were revised with a significant reduction in content.

Japanese teachers have more time during the school day to work together during the school day.

The point is, if we in Georgia intend to be more like the Japanese, we need to pay attention to what the Japanese are actually doing and what their standards mean.

There are good reasons, however, that we should not aim to be exactly like the Japanese. By beginning with a translation of the 1990 Japanese mathematics standards and merely "adopting" them, we lose opportunities to take advantage of and respond to:

changing demands of society and the workplace, particularly the pervasive and growing influence of technology research and curriculum development in the U.S. and internationally, such as the development of algebra in the early grades and the accompanying reconceptualizations of algebra as more than manipulation of symbols the study functions alongside and sometimes before methods for solving equations and techniques of symbol manipulation the importance of data analysis throughout the curriculum, both as important content in itself and to support the learning of other content

It should be clear that strict adherence to poor translations of 14-year-old documents will not serve Georgia's students in this century.

What is Needed

The writers need to work to make the standards documents our own—so that the GPS make sense for and belong to Georgia's teachers, mathematicians, statisticians, and mathematics education researchers, as well as parents and the business community. With the current draft, we expect that neither the Expert Advisory Panel who suggested beginning with the Japanese standards nor the teacher teams who did the writing feel ownership the documents.

The writers must be allowed to deviate from the Japanese model to serve the needs and resources of Georgia. They must elaborate the standards so that they make sense to Georgia's teachers and test writers. Finally, the writing teams should be augmented to include mathematicians, statisticians, and mathematics education researchers to work with the teachers, as each of these groups provides perspectives that the others do not. The GPS should be a consensus document acceptable to all of these communities, and building such a consensus requires significant discussion and development of common understandings.

Too Ambitious Now

Since 1990, the NSF has spent many millions of dollars on the development of new curricula to instantiate NCTM's 1989 standards. These new curricula fulfill many of the goals of the GPS. For example, they take an integrated approach, teaching algebra and geometry throughout the grades, and the process standards, such as reasoning and problem solving, are also incorporated into every unit or chapter.

At the middle grades, several of these curricula include most of traditional Algebra I, focusing on more than the symbol manipulation skills, and often postponing those skills until after the students have considerable experience in relating varying quantities as represented in graphs, tables, real-world contexts, as well as symbols. So in this sense, these new standards-based curricula are roughly a year ahead of the norm in Georgia's schools. When properly implemented, the approach taken in these curricula has been much more successful than traditional Algebra I courses, and we would argue that this change in approach is necessary. Failure rate in H.S. Algebra I is 24% statewide, and more than 40% in some schools. Similarly, many students who take algebra in middle school repeat the course in high school. Thus, moving that course into 8^th grade is not a viable solution, despite the fact that some Georgia counties have taken this approach. For these reasons, we must transcend the rhetoric of "moving Algebra I into middle school." The GPS need to make clear to teachers (and test developers) that the intent is a deeper and richer algebra that is more likely than the standard course to foster learning and understanding, by emphasizing reasoning, problem solving, and multiple representations, and by connecting algebra to geometry and the real world.

Research and experience in the U.S. has shown implementing these new curricula is a long-term process of change, as teachers learn to think about mathematics, teaching, and learning differently. And it is simply not reasonable to expect that teachers can teach more content earlier and more deeply without considerable support from textbooks, peers, and long-term professional development.

Compounding these difficulties of implementation, the draft GPS go a step beyond these new curricula by putting much of 10th grade Geometry into the middle school and putting much computation with fractions and decimals into the 5th grade and earlier. While these may be worthy and reasonable long-term goals, in the short term these goals may be dangerous. It is unlikely, for example, that 6th grade teachers will be able to assume that students already know and understand computation with fractions and decimals.

From another angle, we are quite concerned about teachers in schools who have been working hard to improve but remain "on the list" of schools that fail to meet adequate yearly progress. And now we are going to raise the bar without providing much support. The somewhat gradual implementation plan is not enough to quell their fears. These teachers may feel demoralized; too many of them may just quit.

What is needed

The standards should be high. But the standards must feel within reach, or they will be debilitating to teachers and students. To accomplish this, some of the standards in the current draft should be moved to later grades, at least for the next five years or so.

The solution to the problem of low achievement in Georgia's schools is not merely to push more content into the middle grades. The solution must involve some reconceptualization of algebra, so that what is taught in middle school and high school is not the entrenched course, "Algebra I." This reconceptualization will involve a "balance of concepts and skills," in contrast to the skill-based course that is taught in most classrooms today. If the proposed curricular changes are to be viable, these conceptual changes must be clearly described, elaborated, and illustrate in the GPS.

Lack of Textbooks

At the middle school and high school, there are few texts currently available that would support these standards. U.S. teachers have too little planning time to cut and paste a thoughtful curriculum from other texts, even if they have a broad library of possibilities. Thus, the result will not have sufficient coherence and depth and appropriate breadth.

Because commercial textbook publishers have significant resources and infrastructure and stand to profit handsomely, one can guess that within a few years publishers will introduce new texts "aligned" to the GPS. But in the rush to get something to market, and because Georgia does not have the market clout of California or Texas, these will be reorganizations of existing texts, which often lack depth, pay too little attention to the process standards, and focus mostly on skills. So the result will be texts that not only lack depth and attention to mathematical process but also lack coherence and attention to developing mathematical ideas over time.

An alternative possibility, if the adopted GPS are similar to the draft, is that teachers will try to use available translations of Japanese texts. But these texts are 20 years old, so they lack sufficient attention to data analysis and use of technology. Furthermore, although we found translations of 1984 editions of student texts for grades 7-11, we were unable to find translations of texts for the elementary grades. Neither were we able to find any of the ancillary materials that U.S. teachers expect.

What is needed

The construction of the standards and the implementation plan must pay explicit attention to the availability of appropriate texts. As mentioned above, the NSF-funded curricula would support many of the goals of the new GPS. Furthermore, these curricula include ancillary materials for teachers, and, in most cases, elaborate professional development programs and networks for teachers, schools, and districts adopting the curricula. The state of Georgia would be wise to take advantage of these efforts.

Would it make more sense to introduce the new courses with more flexibility, particularly at the high school level? That way it might be easier for teachers to choose texts and construct courses that make sense for their students. One alternative is to think in terms of units or smaller elective courses.

Process

As we understand the process, neither mathematicians, nor statisticians, nor mathematics education researchers were (or are) among the writers. Instead, the teacher teams tried to carry out the charge of the Expert Advisory Panel to make the GPSs like the Japanese model. (Including university faculty on such an oversight panel in no way substitutes for including them among the writing teams.)

The result of this process is that critical perspectives were missing from the discussion, there was no opportunity for consensus building among various communities, and no one has ownership of the product. The meeting dates for the writers have changed several times. And now, the teachers have been asked to convene from March 8-12 and again in April, perhaps for another week. We know of few teachers who would feel good about calling in substitutes for two weeks in two months, and we expect that some of the writers may have balked. How many of them are willing to do it? Given the state of the current draft, the timeline for adopting the final draft is, simply, ridiculous.

The process for writing the GPS is fundamentally flawed. Because standards are about what is valued, standards must be based on consensus, which requires developing a shared vision and common language. There are at least four constituencies who must be integral to the consensus building and the writing: mathematics teachers, who bring perspectives about the classroom; mathematics education researchers, who bring perspectives about the research on learning, teaching, and teacher learning; mathematicians, who bring perspectives about the mathematics; and statisticians, who bring perspectives about data analysis and applying mathematics to the real world. All of these constituencies have some knowledge of the others, but the quality of the final product and the acceptance of that product by all constituencies depends on the involvement of all four. And there are probably other constituencies (e.g., administrators, parents, legislators) who need some involvement as well.

Consensus building also requires time. Under the current process, with so little time for writing, it is little wonder that the teachers were unable to create a quality product from an incomplete understanding of the Expert Panel's vision.

Feedback

It appears that the Georgia DOEd does not have a viable process for sorting, synthesizing, and responding to the feedback on the drafts. Our understanding is that the comments will be sorted by grade band. If they receive few comments, this may be sufficient. But in the likely event that they receive hundreds (never mind thousands) of comments, they will be completely overwhelmed. Furthermore, they probably will not be able to distinguish between comments of disgruntled mathematicians and parents in California and carefully crafted responses from groups of Georgia teachers, university faculty, or a combination.

We know of many districts, schools, and other groups who plan to submit comments and feedback on their own. We have heard that comments from groups will be taken more seriously than comments from individuals, but right now the Web site doesn't provide a way to indicate that comments are from a group.

What is needed

In an ideal world, we would recommend that the timeline be pushed back a year or more, that the writing teams be augmented to include a breadth of expertise and perspectives, and that they be given ample time and resources to produce a high quality product that is responsive to feedback from the field.

But in case this is not politically feasible, we would like to suggest an alternative approach that fits with Kathy Cox's assertion that these are to be considered "living documents." First, augment the writing teams to include mathematicians, statisticians, and mathematics education researchers. Focus mostly, for now, on sixth grade, as this will be the first to be implemented. Finally, whatever the state Board of Education does in May, they must make a clear commitment of intellectual and fiscal resources to continued development of high quality standards at all grades.

As part of the process, the DOEd has an obligation to demonstrate ways that they have responded to feedback.

The rest of this document is devoted to more specific comments.

Comments on the K-5 standards

We applaud the effort to make K-5 mathematics more meaningful to teachers and children and the attempt to have fewer goals that can be covered more in depth. Not all five NCTM content standards are addressed in every grade level. There is no algebra in K-2 and no reference to children's early algebraic thinking. Data analysis is not a separate strand in K-2 or 3-5 and is a big omission. There are some data analysis activities in the Algebra, Number and Operations, and Measurement Standards. However, to put bits and pieces of it in other strands risks losing the content entirely. Are all Standards to be mastered in the grade level in which they appear? If so, are there any indications of when they need to be introduced? For example, if fractions are not mentioned in K-2, are they to be introduced and mastered in grade 3? The alphabetized bullets under each Standard need clarification. What is their role? Are they examples of what teachers need to accomplish or a new checklist for teachers? The language used in these bullets is many times vague and unclear. For example, what does "distracting appearance" means when referring to numbers in MKN1?

Overall, Kindergarten standards seem not to go far enough. Children in Kindergarten can probably gain more mathematical knowledge than what is expected in these standards. There is a serious problem of alignment between tasks and goals. The tasks need to be looked at more carefully.K-2 and 3-5 Standards need to be better integrated so that there is a clear progression of skill and concept development across K-5, including consistent use of terminology.

Comments on the 8^th grade standards

It looks as though the standards were compiled hastily: Some of the standards seem to be misclassified. For example, Standard M8N1, "the student will develop the ability to represent and analyze mathematical situations using algebraic symbols and to find solutions," seems more of an algebra standard than a numbers standard. In fact, it seems that all of the number standards are really algebra standards. This is probably because algebra is now supposed to be "a study of the relations between varying quantities" rather than manipulation of expressions. The manipulation of expressions is still in the standards, but is now considered part of numbers. It would be more honest to recognize that algebra consists of both a study of the relations between varying quantities and a study of how to manipulate the formulas for these relationships.

Standard M8A1 is "the student will deepen their understanding of different representations of numbers and adequately use numbers in real world situations" seems like it should be more of a numbers standard than an algebra standard. Under "remarks concerning content," applications of similarity are mentioned with reference to standard M8G2.c, but this standard refers to congruence rather than similarity. A careful editing can remedy these problems.

The introduction to the standards emphasizes conceptual understanding, and connections among various math topics, but the statements of the standards seem to emphasize procedural knowledge, and discrete, separate units of knowledge. For example, Standard M8N1 has 3 parts: two of these are completely procedural (a – simplify and evaluate algebraic expressions; c- solve multi-step equations). Part b – translate word phrases to algebraic expressions – is not procedural. However, the expression "word phrases" seems to imply that students will not be presented with a real life problem stated in terms of complete, correct English sentences. The tasks section does present word problems in complete sentences, so the expression "word phrases" should be changed.

The standards discuss similarity under the geometry standards, and proportional reasoning under the algebra standards. Similarity and proportional reasoning are intimately connected, and there should be some tasks to emphasize this. For example, students could be asked to sketch or construct 10 non-congruent but similar right triangles, and for each measure a side (same corresponding side for all the triangles), the perimeter and the area. They could then plot data points: perimeter vs. side length, area vs. side length, etc.

We would be ecstatic if a typical 8^th grader had a solid understanding of the topics covered in the 8^th grade standards, and could perform the sample tasks correctly and with understanding. These standards are setting a very high goal, which is fine, but the high goal only increases the necessity that the standards be written with care. They should provide a useful guide for teachers on how to establish connections among math topics, and on how to combine teaching for both conceptual and procedural knowledge.

The High School Standards

The standards certainly set a very high goal, and it would be wonderful if graduating high school students mastered the topics mentioned. They would indeed be extremely well prepared to enter college. Since the standards set such a high goal, it is vitally important that they be written carefully, and with an eye to being useful for teachers. In particular, the tasks in the standards should be written so as to emphasize problem solving (where problems are by and large real world problems, stated in complete and correct English sentences) and connections between topics.

Nonetheless, the draft Georgia Performance Standards represent a huge leap from current practice in Georgia Schools, and we wonder whether the goal is attainable in the near term or whether it is wise to make such drastic changes all at once. Some of the proposed content is not necessary for college intending students, though it may be nice for students who intend to major in the mathematical sciences. Still, Math IV could be combined into Math III and Precalculus because it won't take a year.

We do not think it is realistic to expect to be able both to cover topics in greater depth and with greater conceptual understanding, and to also cover more topics. The standards have a number of topics (e.g., sequences and series, induction, binomial theorem, complex numbers, matrix computation, parametric equations, polar coordinates, vectors, DeMoivre's theorem), which some UGA mathematicians never had in high school. As a comparison, we used to teach polar coordinates in the second quarter calculus course at UGA, and sequences and series in the third quarter calculus course. We now teach polar coordinates in the third semester calculus course, and sequences and series as a separate course in its own right. We are concerned about the feasibility of covering all the topics in the standards in depth. It may be a good idea to introduce students briefly to a topic, just so they will have heard about it. But the standards, if they are to be useful guides for teachers, should be clear about what is supposed to be covered in depth, and what can be mentioned superficially.

Apparently Missing Content. Because the standards are so brief and lacking detail, they give the impression that much important mathematics content is missing. For example, many of us were concerned initially that Euclidean Geometry was gone. Finding parts of it in 7^th and 8^th grade standards did not give a broad or deep enough sense of the content, and, in particular, did not give us confidence that students would have sufficient sense of proof. Furthermore, pushing Euclidean plane geometry into 8^th grade seems to be expecting a lot of both teachers and students.

Texts. What texts will teachers use? Mathematics teaching in this country is dominated and guided by textbooks. With such careful prescription of content at every grade, there will be considerable difficulty finding texts that align with the standards. Available translations of the Japanese texts align well, at least for grades 7-10, but these were written for different students and different teachers, and they are based on Japanese texts that were written 20 years ago.

Implementation. What support is going to be provided for teachers to implement these standards? We have serious concerns that teachers will need considerable help putting these standards into effect. This need will not be fulfilled by a two-day workshop in the summer, particularly if only one teacher per school attends this workshop and is expected to "disseminate" the ideas to her or his colleagues.

Integration. An integrated approach is a good idea, but how to the standards for a course tie together? The current approach (a year of algebra, a year of geometry, and a year of algebra) needs improvement, but the current draft suggests we might get two months of geometry followed by two months of algebra, and so on. The connections between numbers, algebra, and geometry need to be made early and often, but unfortunately the Japanese texts do not appear to make these connections explicit. And although we applaud the decision to stop teaching the same topics over and over, we are concerned that students may not retain much of what they have learned without some repetition. One solution to both the lack of integration and the problem of retention is to use old ideas regularly in the service of new mathematics. This approach would need to be explicit and well illustrated in the standards.

Process Standards. The process standards sound lofty. For example, students will "develop an ability to evaluate mathematical arguments." But in the current draft it is not at all clear how students will learn to do this. Here are ways that the process standards should clearly and explicitly be threaded throughout the content standards:

Students should be setting up word problems in every standard and frequently using the mathematics to model real world problems.

Students should often be given opportunities to solve problems before they have been told how to solve them. Students need to spend a lot of a time on problem solving, giving them the feeling that mathematics make sense.

Students should be encouraged to explain their reasoning both verbally and in writing (with complete sentences) in every standard.

What about students who have no intention of going to college? The idea that they might decide to go to college 10 or 20 years in the future is not a motivational for some students. Reducing math phobia and preparing them for life might be more reasonable and worthwhile goals.

This draft needs considerable work. If the goal is to prepare students for college, it is not reasonable to ask teachers to do this independent of college faculty. We would recommend that teachers, mathematicians, statisticians, and mathematics education researchers work together to build common understandings and consensus about what students should know and be able to do. We would recommend that the Department of Education support week-long workshops to work out some of the details.

The Math I-IV sequence

With much of the content of traditional Algebra I and 10^th-grade Geometry pushed back to 7^th and 8^th grade, students in the standard track should be taking Precalculus in their junior or senior year, depending upon how much probability, statistics, and discrete mathematics content are included in Math II and III. In other words, it appears that Math III and Math IV are relatively thin on content.

At the same time, there a conflict between the perceived need to take mathematics every year in high school (for college admission requirements) and the intent of getting all students ready for college, the workplace, or everyday life. Other than exponential functions, it is hard to argue that any of the content currently in Math III, Math IV, and Precalculus is necessary for college, except for those students who are intending to pursue mathematics-intensive careers. And even for those students, not all of the content is necessary, as described above. Regarding preparation for everyday life, some of the content of Math I and II might even be unnecessary. Interestingly, both the state of Michigan and the Achieve partnership have avoided these problems by developing standards only through the eighth grade, at least for the time being.

If we really take the Japanese model seriously, then we should offer several optional courses in 11^th and 12^th grade. AP Calculus and AP Statistics clearly are two such courses. Other possible courses might be Precalculus, a topics course for students intending on pursuing mathematics-intensive careers, or other topics course for students with other career aspirations. At UGA Mathematical Modeling and Mathematics of Decision Making are examples of such courses.

A suggested alternative to Math I-IV

In keeping with the comments above, and in an attempt to stick somewhat closely to the Japanese model, the table below shows the placement of topics in the 1984 Japanese texts, the 1990 Japanese standards, the draft GPS, and a suggested alternative.

	Japan 1984	Japan 1990	Draft GPS	Suggested GPS
Square Roots	Grade 9	Grade 9	Math I	Math I
Polynomials	Grade 9	Grade 9	Math I	Math I
Quadratic Equations	Grade 9	Grade 9	Math I	Math I
Functions (simple and reps.)	Grade 9	Grade 9	Math I	Math I
Circles	Grade 9	Grade 9	Math I	Math I
Similarity, Pythagorean Theorem	Grade 9	Grade 9	Math I	Math I
Probability, sampling	Grade 9	Grade 9	Math I	Math I
Quadratic Functions	Math 1	Math I	Math II	Math II
Triangular trigonometry	Math 1	Math I	Math II	Math II
Counting, sequences, permutations, combinations	Math 2	Math I	Math II	Math II
Probability (indep., complem., exclusive, expected value)	Math 2,	Math I	Math II	Math II
Plane Geometry (Analytic)	Math 1	Math A	??	Math II
Computation and Computer	Math 2	Math A	??	Math II
Exponential functions	Math 2, BA	Math II	Math III	Math II
Sequences & Series, Induction, Binomial Theorem	Math 2, BA	Math A	Math II	Math X