"[The State Board of Education's Math Standards] are demanding...and students who meet these standards would be well prepared for the application of mathematics and for further study of mathematics.

"The editing and re-writing has resulted in the near-elimination of jargon and unclear usage of words....The statements for the most part are explicit and would provide sufficient guidance for the application of the concepts or operations being described. The 8-12 standards rely on accepted mathematical terminology... For the most part,...the standards are pleasantly devoid of jargon and statements that are difficult to interpret. They also appear to cover the essential concepts and operations in mathematics, both at the K-7 and 8-12 levels.

"The California Board of Education made the right move on math standards. Teachers and students need to focus on math basics, learned through memorization, drill, and practice. Once students have mastered the basics, going on to grasp broader concepts is not a problem.

"Unless the basics are addressed, the current labor shortage facing the electronics industry will only get worse. Two-thirds of American Electronics Association-member companies that bring in less than $20 million a year in revenue are dying because they can't find the people to do the jobs."

"Californians tolerate terrible inefficiencies in the management of their schools--inefficiencies they would never tolerate as customers of any company. . .I am one Californian who fully supports the new math standards."

"In the next decade, the American businesses will want to hire over a million new computer professionals. But at this point our schools are not preparing enough people to fill those jobs.

"The new California Math Standards approved by the State Board of Education are solid on the content of mathematics. Students will learn arithmetic skills; students will master algebra and geometry--which are the doors to good jobs and further education. The standards are strong on applications, conceptual understanding and mathematical reasoning, which will help students after they finish school in performing well on the job, particularly in high-tech companies. If our schools, teachers, and students will live up to these new math standards, the students educated under them will be ready for the jobs of tomorrow."

"It is now more important than ever for California schools to produce well-educated graduates with the skills to meet the job demands of the new knowledge-based economy. High standards in math and science are not optional if we are to fill the more than 346,000 high-paying information technology jobs now vacant across the U.S.

"The Third International Math and Science Study (TIMSS) showed that American students do not measure up to the global competition. While we may not be feeling the repercussions yet, today's 'Internet Generation' will soon. The Internet will enable companies to locate employees anywhere in the world. Jobs will naturally migrate to those countries and regions that have a

well-educated workforce.

"In business, we apply standards and test for results. The same must be done in education. Without a system of standards and testing that fully discloses both the strengths and weaknesses in our education system, we will not be able to improve education in California. Because California's new math standards are high, uniform, and achievable, they provide a mechanism for attaining excellence.

"Cisco is sponsoring a program called Cisco Networking Academies that is helping to prepare today's students for tomorrow's jobs, but its success in part depends on our students' abilities to excel in math and science. I believe California's new Math Standards will help build the skills that students will need to compete in today's global economy."

"Our students need the fundamentals of arithmetic, algebra, and geometry to form the necessary foundation for understanding mathematics. And teaching these basics still leaves plenty of room for creative approaches to understanding abstract math concepts. Neither approach precludes the other--we can have both. The new state math standards call for teaching both the basics and the concepts. Locally, we need to tell our schools to get on with it! Every week of delay is another week of random experimentation on our children. Mathematics is not a social science, and education should not be a political science."

"As a technology company director as well as the mother of two small children, I've become increasingly alarmed at the lack of precision, inability to perform fundamental math calculations, and general 'fuzzy' understanding of mathematics exhibited by young recruits out of the California school system.

"We live in a world in which a solid ability to perform mathematical operations coupled with a firm conceptual understanding is essential to most of the disciplines and careers our young people will pursue. While I'm a big believer in case studies, visualization, and real-world examples, these must be used to aid in building a firm foundation of mathematical understanding and ability, not put in place of it.

"The new mathematics standards for California's public schools adopted by the State Board of Education are, in my opinion, fundamental and essential to the building of the skills necessary for our children to compete in and contribute to the economy of the next century."

"[T]he American Federation of Teachers has reviewed the 'new' standards the [State] Board approved [on Dec. 11, 1997]. Our judgments are based on the criteria used in our report Making Standards Matter, which looks at the clarity and specificity of academic standards.

"Based on these criteria, we consider the math standards strong enough to lead to a common core of learning. The standards are written clearly, are quite specific, and are firmly grounded in content. This is especially true for the standards from grades 5 through high school."

"....When I read the Board's document, I see a call for both conceptual understanding and fluency in technical skills.

"Some people believe that if students have conceptual understanding alone, they can solve problems they have not seen before. This was the philosophy of the New Math of the 1960s. Unfortunately, this does not work for most students. In addition to understanding--whether 'conceptual' or a more serious kind of understanding--students need to learn technical skills and have to have considerable experience doing multi step problems."

"....The needed revision has now been supplied by the Board. It can be argued that some other delicate educational issues were raised in the process, but these should have been handled in a scholarly manner without the hue and cry. On the issues of mathematical illiteracy and incoherence, however, there is no question that the revision is a success."

"....It is perhaps the Post-7 section [of the Board's revised draft of the Standards] that has generated the most heat among mathematics educators in California, for whereas the original draft prescribed an 'integrated curriculum,' the Board's present version offers two options. Its standards describe the high school material by subject matter, 'algebra,' 'geometry,' etc., permitting (and maybe encouraging) schools to offer them as separate courses with these subject

names, but also, if they wish, to repackage the same material, or major parts of it, in 'integrated' form, calling the resulting courses Math I, Math II, and Math III."...

"It is ludicrous to suppose that the Stanford mathematicians have the intention of reducing mathematics education to the kind of drill in mindless algorithms their opponents paint them as advocating. Above all else, mathematicians want the schools to teach that which generates understanding and not memorization of routines. Above all persons, mathematicians understand the implications of lessons that affect to do the one or the other...

"[I]t is flat-out false that the revision has as its purpose the 'return to [rote-learning] basics.' This is only its characterization by mathematically illiterate people. . ."

"The set of standards which has been approved by the School Board states in clear and unambiguous language the mathematics content which should be mastered by all students in our schools. They are content standards as they should be. The combination of these standards together with the proposed Framework will provide for the balance between conceptual understanding and computational skills which was called for in the 1996 Program Advisory. The standards are moving in the direction of the mathematical level which Japanese students are expected to attain and should help our students' international standing if properly implemented in the classroom. The standards do allow for an integration of topics, but do not mandate it. It is important that an integrated curriculum treat topics in sufficient depth, and not become an unfocused hodgepodge of shallow topics covered without a clear goal. Some recent attempts at

integration of topics are of the latter type, and we need to avoid these in our classrooms."

"The California Board of Education is to be congratulated for developing a set of K-12 standards in which expectations are high, yet reasonable. These standards give teachers a well-organized summary of the content needed in a rigorous and solid mathematics program. It is good to see a document devoid of jargon. Its clear, concise, and precise language will enable teachers to know the goals students must attain to advance in their understanding of mathematics."

"As one who has been a Mathematics Professor for 30 years and is at present the parent of two teenage children, I would like to say how delighted I am by the standards recently put forth by the State Board of Education. As a professor, I have been appalled by the lack of mathematical preparation evidenced by my students. As a parent, I have been able to understand better how this wretched state of affairs has come about. One of my daughters has been an honors Math student in the public schools in Santa Monica. However, I recall with horror how, in the 7th grade, she could not multiply a number by 10 or add 1/2 to 1/4 without the use of a calculator. I also recall my puzzlement at how in the earlier grades most of her arithmetic homework consisted of estimating answers and relatively little work was devoted to getting an

exact answer.

"....What I find represented in these standards constitutes what our students need to know and understand. To attain this knowledge and understanding will require both effort and imagination on the part of our students. I wish to reiterate that I feel hopeful that the children and young people of California will have a chance, for the first time in years, to perform well in the honest and noble undertaking that is Mathematics."

"The Mathematics Standards approved by the State Board of Education require both genuine conceptual understanding of mathematics and the requisite facility with basic skills... California's newly adopted standards for mathematics set a high benchmark and are refreshingly clear and easy to understand. They leave pedagogical issues to the classroom teachers..."

"Most important about this formulation of the standards is its specificity. Rather than relying on vague phrases and good intentions that have dominated mathematics education for the last 10 years, these standards specify the mathematics that is to be covered."

"We who teach mathematics at the college level have noticed in recent years not only a decline in arithmetic and algebraic skills on the part of our students, but also a decline in the ability to analyze simple problems, to say nothing of complex ones.

"It is the twin decline in both skills and problem-solving ability that has created the demand for the revision in the standards. Indeed, we who teach mathematics believe that it is impossible to develop good problem-solving abilities without a solid foundation in arithmetic and algebra. Once that foundation is in place, the revised standards then build on this foundation to improve the ability to solve problems. Those who claim that the revised standards neglect the development of problem-solving skills are misleading the public."

"As a mathematician I am delighted by the [Board's] draft of the standards. The format is clear, concise, and extremely readable. The level of expectations is high, as is appropriate. The mathematical language is precise and leaves little room for doubts about what students will be required to learn. I am particularly pleased with the format of the 8-12 standards. Dividing the subjects by discipline is by far the most sensible approach. It allows us to see clearly what topics are supposed to be covered within each discipline, while allowing individual school districts the flexibility to implement an integrated program should they wish. The standards commission's previous draft, mandating an integrated curriculum for grades 8-12, was highly experimental. It would have been disastrous to approve it.

"....As [the mother of three children, two of whom are in the public schools] I am also delighted....I look forward to seeing my children thrive in mathematics. I believe all children will benefit from the clear and high expectations represented in this document."

"Although there is much to be learned about how children understand and acquire mathematical concepts and procedures, what is known suggests that children's learning will be greatly facilitated with the implementation of a highly organized, rigorous and explicit curriculum. On these dimensions, the standards recently adopted in California are the best that I have seen

in this country."

"These new math standards are good because they expect students to give it their best. They ask students to dare to master math. They are clear and precise in saying what math is and what schools should expect from students. They are good because they discourage Mickey Mouse math classes with no content. Mickey Mouse classes don't prepare students for life. They don't prepare students for the future.

"We have to increase our expectations. We have to shoot for the highest standards if we are going to compete with Japan and other industrialized nations. If American students are prepared in math, they can innovate, they can anticipate, and America will succeed in the world."

"....The 'reformers' will, no doubt, characterize the State Board's post-seventh grade standards as 'traditional.' We must not be put on the defensive by this characterization. It means that these standards contain elements which enable them to stand the tests imposed by time and experience. It does not mean that these standards foster 'back to basics' efforts to emphasize the assimilation of facts at the expense of concept building. These standards are balanced because they require the student to learn the facts that he needs as a basis for the high level of mathematical reasoning demanded in every section of these outlines.

"Again, I commend the State Board for successfully demanding standards for mathematics which will serve to raise the level of student achievement throughout the state and will also enable them to cope with the manifold deficiencies of the 'reform' programs that are now sweeping the nation. If these California State Board Standards receive the publicity they deserve, they may enable other states to do the same."

"As a former mentor teacher in the California Math Project, I believe in a balanced mathematics curriculum from the basics up. Therefore, the action the California Board of Education has taken to clarify and specify exactly what our students must study and master is long overdue and welcome to those of us in the trenches...

"Teaching [as I do] in an inner city Los Angeles high school, these precise, rigorous standards will give the opportunity to implement real student, teacher, and administrator accountability. Precise, content based standards are accessible to 'study,' the long forgotten word in American

education. And a system where students are held accountable, working somewhere near their potential, is one that will produce the educated populace our state needs. Depending on radical curricular soufflés to solve all problems is utopian and delusional."

"The standards adopted by the California State Board of Education are truly outstanding. These standards will provide teachers, students, parents, and the community with absolute clarity when evaluating programs and individual achievement. Since they are neutral with respect to pedagogy, educators are being provided with a wide latitude in selecting programs which best address their students' needs. The return to the development of basic skills and avoidance of calculator dependency will significantly enhance students' opportunities to develop conceptual understanding. Since the standards advocate a 'knowledge based' developmental program, students will be able to confidently advance through a challenging mathematics program.

"I am both a high school classroom teacher and a department chairman. Teachers throughout our department share my enthusiasm for the 8-12 standards--because of their emphasis on proof, their thoroughness in addressing the concepts, and the clarity that resulted from organizing the standards by disciplines."

"I am writing to congratulate you [the State Board of Education] and offer my support for the revised math standards [the Board has] created....[T]he standards you have set are consistent with what the most reliable scientific research tells us about the soundest principles to follow in teaching early math....[B]oth controlled psychological studies and large system-wide comparisons...converge on the principle that early mastery of computational skills is the only secure route to higher-order understanding and problem-solving by all students."

"As an admission officer responsible for reviewing high school curricula in math and science for proper college preparation, I am especially heartened by the new standards adopted by the California State Board of Education. They provide a sound basis for students to be seriously challenged in high school and to acquire the greatest depth in mathematics prior to attending college. This is especially crucial in a technologically-driven state like California where mathe- matically-advanced students can use a college or university to pursue work in greater depth."

"The initiative to foster basic mathematics in K-12 education exemplified in the new California Math Standards is a welcome change from the quagmire of 'new-new math,' 'fuzzy math,' and 'rain forest ethno-mathematics' that has plagued California's students in the last decade. Rigor in mathematics is the driving force for economic growth and is fundamental to our future success. Raise the bar, and all students will successfully jump over it as long as we have teachers who are content experts in math and can really teach that content."

"To believe that tomorrow's world of industry and commerce does not require facility with arithmetic basic skills indicates lack of experience in today's world. Calculators are not always available and on-the-spot calculations or very rapid approximations are of the essence in formal and informal scientific discussions or the fast moving world of Wall Street. These skills come only from repeated practice of those skills with sufficient understanding to devise shortcuts and reformulations in one's head. No one would suggest that a golfer could become competent without extensive practice in basic skills, how could it be different with mathematics?

"Obviously these skills, though necessary, are not sufficient; broad problem solving and analytical reasoning are also of the essence, but to suggest the latter can be taught without a grounding in the basics is absurd."

"The [California] Board of Education's draft standards [are] a breath of fresh air in the standards business. The Post-7 standards follow standard math curricula, as [they] should....

"[T]he K-7 standards are sound and clear. The detailed specificity will help teachers focus on essentials....There is a coherence to the material....

"I appreciate the Board of Education not pushing any particular teaching strategy. Education has been too burdened by 'revolutionary' doctrines that don't work."

"The changes to the California Math Standards made by the California Board of Education are a great improvement. The precision has been greatly increased -- a help to textbook authors and test makers. The new standards have been changed to define the desired goal rather than

the process. That is, they define what a child should know and be able to do. Previously, they specified how a child should learn a concept. Now they allow teachers to achieve each goal in a way that is best for each child."

"....In contrast, the standards produced by the California State Board of Education are something Californians can be proud of. They are completely devoid of jargon and clearly state specific skills that the students must master. The emphasis in the later grades on mathematical proofs is especially welcome since we live in a time when logical rigor is neglected all too often. A student who can do what the State Board's standards call for will be well-prepared for engineering programs in higher education and engineering jobs in later life."

"[Mathematics standards are] critical to our nation's economic future. . .Many students in [California] are currently taught mathematics without textbooks using a 'brainteaser of the week' approach. Rather than learning to solve problems in a methodical and logical fashion, they often perform politically correct exercises in creative writing without clear answers ('What did you learn about yourself today?').

"The study of mathematics and science teaches important skills in critical thinking and problem solving in addition to helping us understand our increasingly technical world. The 'investigative' methods advocated by proponents of the 'new new math' can be useful in stimulating a student's interest, but can never replace the need for a mastery of basic skills and facts. . .I for one would like my airplanes, bridges, polio vaccines, federal budgets, and school curricula designed by people who understand the value of correct answers and who possess the logic to arrive at them."

"....The Board's draft has the additional major improvement that it concentrates on what students should know and do--and not on lobbying for one teaching method over others."

"....The board-modified standards clearly define what students should learn, while letting local districts decide how to teach.

"In addition, the board reformulated Algebra I, Geometry and Algebra II, protecting our students from the Commission's experiment with 'integrated' math. The Commission's recommendations for grades 8-12 mixed topics from algebra, geometry and trigonometry, and stretched teaching of a single subject over four to five years; Instead of teaching all of Algebra I in eighth grade, some beginning topics would have been pushed back to grades 9, 10 or beyond."

"Let's go beyond the philosophical debate between the mathematics traditionalists and the mathematics reformers. What is important is what is best for all students, to learn what we want them to know to be able to compete. We cannot fail. They have only one journey through our schools.

"The 1995 State Mathematics Task Force called for rigorous standards. The State Board of Education has responded to this call and approved rigorous standards. To reach those high standards, a variety of teaching methods is required. The single-method approach of the past is too narrow for the current needs of our diverse students. However, this decision should be left to local school boards. Fortunately, the State Board's decisions have given us the best combination: high, rigorous standards and flexibility of approach for local districts.

"Students need to master the basics to be able to calculate correctly and grasp the more challenging and complicated mathematics. The new mathematics standards provide for both."

"I feel confident that the new math standards will give students a solid base of math skills and knowledge. I believe that only with such a strong foundation will kids be able to understand and work with more sophisticated and difficult math. The standards are rigorous and comprehensive, clear and precise. Our kids will be challenged--which is good! Students are capable of much more than we are presently asking of them. This is my personal opinion as an active parent and not that of any of the groups in which I am active."

"As a newly elected trustee and the mother of two children, I would like to thank the State Board of Education for establishing math standards that are clear, concise, and rigorous. These standards offer the children of California goals required for effective competition in our global economy. They also allow parents to compare grade level expectations to their own children's classroom work.

"My district is in the middle of a curriculum adoption for Mathematics, and we have spent several years working on our own district benchmarks. It has been a good test to compare our own expectations to the new standards. Our concern is not that the standards have been 'dumbed down' as some would have us believe. Instead, we recognize that these standards are rigorous and demanding. Our teachers will require significant staff development, and our students will require considerable support for us to reach the goals set forth in the math standards."

"Because the Math Standards are clearly measurable, they give parents a valuable tool with which to evaluate the math programs in their children's schools. Parents are then in a position to go to their local school boards and give kudos or make demands--and woe to the elected school board member who fails to listen."

"Having been hired by major corporations such as Motorola, at some expense to those companies, to train high-tech workers in basic computation and problem solving skills not learned in school, we have seen first hand the need for solid basic computational skills in a high-tech world. The issue is not whether individuals need to be able to compute rather than be able to solve problems and reason about math, it is that both are needed and are not mutually exclusive.

"We believe that students cannot learn higher level math without being facile in basic arithmetic skills. For example, factoring equations whether in algebra or calculus requires familiarity with basic multiplication and division in order to readily see which number combinations can be used as factors. More important is the fact that teaching arithmetic is in no way incompatible with teaching the analytical ability needed to analyze problems and apply mathematical modeling--and these basic skills are also necessary [in and of themselves]."

"I want to thank the State Board of Education for the December 11 revisions to the math standards. I have seen both the K-7 standards and the 8-12 standards and find them to be much more clear and concise, with a needed emphasis on basic skills.

"I am an industrial arts teacher as well as a math teacher and have watched the students become less and less able to perform in a technical-industrial setting because of a lack of basic math skills. Fully half of my freshman drafting students are unable to use a drafting scale because they do not know their fractions. I must give them a diagram of an inch with the various fractional divisions labeled so that they can perform the most basic measuring tasks. I am also heartened to see the requirement that students be able to perform basic calculations without a calculator. The overuse of calculators has resulted in a near total loss of number sense.

"We see many high school students who after making a simple mistake such as getting 60 as an answer to 20 times 30 cannot recognize that 60 is not a reasonable answer. Or that 8 is not a reasonable answer to 1/2 times 1/4. We all make those types of mistakes from time to time. The difference is that those of us who had considerable practice between the first and eighth grades can usually see when we have an unreasonable answer."

"California's Board of Education has backed new math standards that stress knowing ''times tables'' and mathematical formulas. That's good news, not only for California students, but for kids all over the country....

"Unfortunately, the state has [in the past] produced some fuzzy-headed thinking about how kids should learn. In math instruction, right answers have taken a back seat to learning 'the process' of solving problems. Teachers have used role-playing, essays and group work to encourage kids' love of math. And other states have followed suit.

"But now California's state school board has come around to the fact that right answers to math problems are important. And it's realized that right answers require basic skills, such as knowing how to add, subtract, multiply and divide.

"With that in mind, the board this week adopted standards that require a third-grader to know the multiplication tables for the numbers one through ten. And fifth-graders should be able to do problems with decimals and negative numbers....

"To be sure, learning how to solve problems is important and should be taught. But only if basic math skills are taught first. If students don't have these basic skills, how will they know if they've actually solved a problem?

"...[T]he board should not relent. The future of kids across the nation is at stake."

"The State Board of Education's proposed math standards correctly emphasize a return to the basics that every California student needs to know without depending on a calculator. The fundamental skills--adding, subtracting, multiplying and dividing--form the building blocks of a

solid math foundation. Youngsters need to know, yes, memorize, certain facts. Of course in a more sophisticated world they need to do more; they need to be able to conceptualize and use all the technology available. But the state's outrageously dismal math scores show that students can't get there without the road map of basic skills."

"On Monday, the State Board of Education adopted tough new mathematics standards for California's public school children in kindergarten through eighth grade....

"[A] major reason for adopting new standards is precisely to get past what has proved to be, in large measure, fruitless experimentation that has gone on in California schools for two decades or more....

"It seems clear that these changes [the Board's revisions] have made the standards more precise, not less. The Board of Education should stay the course with its revisions."

"....The board appears to be trying to keep the state from lurching once again into an experiment with the education of 5 million schoolchildren. If it can accomplish that, without lurching too far in a fundamentalist direction, the board will have done California a favor. This state has seen far more than its share of educational course corrections; it would be nice, for a change, to see the ship steered somewhat straight."

"Earlier this week, the State Board of Education endorsed tough math standards that emphasize the basics for kindergarten through the seventh grade....

"The tentative accord on math standards is especially encouraging because it underscores the importance of learning the basics. Ralph Cohen, a Stanford math professor who helped draft the tougher standards, put it best when he said teachers should not expect children to understand concepts without first gaining a solid grounding in the fundamentals.

"Although critics complain such grounding kills student interest in math, the fact is that kids who are deficient in the basics are being set up to fail. Granted, they may not enjoy mastering the multiplication tables and learning how to borrow and carry while adding and subtracting. But these are the tools that will enable them to function in society, because their calculator may not always be at hand. Indeed, the new standards discourage the use of calculators on state tests....

"California cannot afford to subsidize a public education system that expects so little from its students. That is why the bar must be raised on academic standards and student achievement."

"The State Board of Education did the right thing last December when it unanimously adopted math standards that emphasize the basics for kindergarten through seventh grade.

"Since then, however, the board has been attacked by critics who contend that requiring students to master the multiplication tables, decipher fractions and learn other math fundamentals is the surest way to kill off their interest. Better, they argue, to allow students to brainstorm in the classroom, to ask questions of one another and otherwise engage in abstract exercises to solve

math's mysteries.

"To its credit, the state board rejected this fuzzy thinking in favor of a more traditional approach to math instruction. That should have been the end of the story, but the critics have mounted a fierce counterattack.

"Before adopting the more traditional math standards, the state board insisted that they be firmly grounded in solid research. That research clearly shows that students who lack the fundamentals are destined to become confused and discouraged as they try to fathom more advanced concepts.

"Students may not like putting forth the effort it takes to master the basics. But that's no excuse for educators to shortchange kids for fear of offending them."