The Honorable the Members of the Board of Regents


James A. Kadamus


Full Board


Report of the Mathematics Standards Committee


October 27, 2004




Implementation of Regents Policy


Goals 1 and 2






The Mathematics Standards Committee was formed in response to the recommendations made to the Board of Regents by the Independent Math A Panel which investigated the high failure rates on the June 2003 Math A Regents examination.  The Commissioner's charge to the Committee was to examine the existing Regents learning standards in mathematics, consider relevant research and other standards from the U.S. and other nations, and then propose modifications to the Regents mathematics standards to improve clarity, specificity, and functionality.


The Committee has completed its charge under the leadership of the Co-Chairpersons William Brosnan and Theresa McSweeney.  They will present the Committee's report at your November meeting.  Attached is the report which consists of:  (1) an overview of the recommendations and the rationale for changes; (2) Attachment A which identifies the current learning standards and key ideas for mathematics, science and technology; and (3) Attachment B which identifies the proposed content and process standards for mathematics as well as grade-by-grade performance indicators.


            We propose that the Board concur with the Committee's recommendation for a public comment period and that the public comment period begin now and conclude in mid-December.  If there is a need, we will reconvene the Committee to modify the recommendations based on public comment.  We will bring the final report to the Board in January and will propose that the Board take action to approve the revised learning standards for mathematics.



Mathematics Standards Committee


Report to the New York State Commissioner of Education


November 4, 2004



Sherri Blais

   Teacher of Mathematics

   Monticello School District

Carlos X. Leal

   Elementary Math Lead Teacher

   Rochester School District

Judith Blood

   Elementary Teacher

   Ithaca School District

Jennifer Lorio

   Elementary Teacher

   Yonkers School District

James Boswell

   Alternative Education Teacher

   Capital Region BOCES

Gwen McKinnon

   Middle School Principal

   Syracuse School District

William Brosnan, Co-Chairperson

   Superintendent of Schools

   Northport-East Northport School District

Theresa McSweeney, Co-Chairperson

   Teacher of Mathematics

   Marcellus School District

Jacqueline Bull

   Coordinator of Mathematics, K-8

   Clarence School District

Brenda Myers

   Deputy Superintendent

   Broome-Tioga BOCES

Melba Campbell

   Teacher of Mathematics

   Samuel Gompers High School (NYC)

Miguelina Ortiz

   Elementary Teacher

   Baldwin School District

William Caroscio

   Teacher of Mathematics

   Elmira School District

Alfred Posamentier

   Dean, School of Education, City College

   Professor of Mathematics

Vincent Cullen

   Certified Public Accountant

   Long Island

Roderick Sherman

   Teacher of Mathematics

   Plattsburgh School District

Andrew Giordano

   Construction Engineer


Susan Solomonik

   Teacher of Mathematics

   IS 119 (NYC)

Carolyn Goldberg

   Professor of Mathematics

   Niagara County Community College

Debra Sykes

   Director of Mathematics

   Buffalo School District

Robert Gyles

   Professor of Mathematics Education

   CUNY Hunter (NYC)

Thomas Tucker

   Professor of Mathematics

   Colgate University, Hamilton

Daniel Jaye

   Assistant Principal/Math Teacher

   Stuyvesant High School (NYC)

Stephen West

   Professor of Mathematics

   SUNY Geneseo








The Committee would like to express its appreciation to several members of the State Education Department who were extraordinarily helpful, including Deputy Commissioner James Kadamus, Associate Commissioner Thomas Sheldon, Assistant Commissioner Jean Stevens, Assistant Director of Curriculum, Instruction and Instructional Technology Anne Schiano, Mathematics Specialists Teri Calabrese-Gray and Michelle Kline, Administrative Assistant Judi Golombiski, and Secretary to the Associate Commissioner Maryann Jansen.  We were very impressed by their commitment and dedication to the public schools of New York State, and to those for whom we are all here, the children.

Table of Contents




I.        Introduction........................................................................................................................... 1

          A.      Background................................................................................................................ 1

          B.      Charge to the Committee......................................................................................... 3           

          C.      Committee’s Response to Its Charge..................................................................... 4           

II.        Important Points Relative to Our Work.............................................................................. 5

          A.      Guiding Principles..................................................................................................... 5           

          B.      Time Constraints....................................................................................................... 6           

          C.      Field Review of Our Work is Recommended......................................................... 6           

          E.      The Importance of "The Power and Beauty of Mathematics"............................... 6           

III.       Recommendations.............................................................................................................. 8           

          A.      The Standards........................................................................................................... 8

                    1.       MST Standard 3.............................................................................................. 8

                    2.       Key Ideas.......................................................................................................... 8

                    3.       Detailed Performance Indicators................................................................... 9

                    4.       Implementation Timeline for 3-8 and High School..................................... 10

          B.      High School Mathematics Program, and Graduation Requirements................ 10

                    1.       Graduation Requirements and Regents Examinations............................. 10

                    2.       Course Content and Labels......................................................................... 11

                    3.       Additional High School Program Issues..................................................... 13

          C.      Guidance to Classroom Teachers on the Standards.......................................... 14

          D.      Technology and Mathematics Instruction.............................................................. 14

          E.      A Capacity Issue – Adequately Prepared Teachers........................................... 16

IV.      Summary of Changes to Performance Indicators.......................................................... 17

V.      Summary............................................................................................................................ 19


Attachment A.  Current MST Standard 3 and Seven Key ideas

Attachment B.  Proposed Mathematics Standard, Content Strands, Process Strands, and Grade-By-Grade Performance Indicators (PDF File) (Word File)






A.        Background


This Mathematics Standards Committee was formed by the New York State Education Department (SED) in response to a recommendation made to the Board of Regents in October, 2003 by the Independent Math A Panel, a group formed to investigate very high failure rates on the June, 2003 Math A Regents Examination.  The Panel’s findings and recommendations included the following in the Executive Summary (p. 4):


Finding  1:        The Math A standards lack clarity and specificity.


Recommendation 1A:             Educationally useful standards must be developed in mathematics for each grade, K-8, and for Math A and Math B, that consist of a clear, well-defined set of skills, the mastery of which is demonstrable.


Recommendation 1B:             SED should establish a mathematics standards committee to rewrite the standards into functional form, and to meet regularly in the future to analyze test results, thus ensuring continuous relevance.


Recommendation 1C:             SED should develop and disseminate suggested curricula for mathematics instruction for each grade K-8, and for Math A and Math B (p. 19).


Recommendation 1D:             To benefit from the extensive research and deliberation of the current Math A Panel, some of the current Panel members should be included in both new committees recommended in this report, i.e., the standards committee, and the curriculum development committee.


The establishment of this Committee was in response to the Panel's Recommendation 1B.  In accordance with Recommendation 1D, several members of the original Panel were appointed to this Committee, including both Co-Chairpersons. The Panel report also stated:


This committee should include a large cross section of adults including mathematics teachers, university mathematicians, professors of mathematics education, special education teachers, parents, and adults who work with mathematics in real work applications, both in the professions (for example, engineers and accountants) and in the trades (for example, carpenters and electricians).  The Panel envisions that this group would meet at least once a year to review the exams against the standards, in order to provide continuity over time. (Page 19.)


SED's efforts to include a large cross section of adults was, for the most part, successful.  The Committee included mathematics educators from a wide range of grade levels, and from a wide range school district types and geographic areas.  The Committee also included an accountant, an engineer and, as a tradesperson was not available, a BOCES teacher of occupational education, with a background in mathematics.  In addition, the Committee included representatives from higher education, who are knowledgeable about mathematics, mathematics education, and the pre-service and in-service preparation of teachers.  The Committee commends SED for this broad effort, and believes this diversity was critically important to our efforts.


The Committee held 20 days of meetings, all in Albany:


January 8                                           January 29                                         January 30

February 26                                       February 27                                       March 16

March 17                                            April 29                                               April 30

May 24                                               May 25                                               June 29

June 30                                              July 19                                                July 20

July 21                                                August 9                                             August 10

September 27                                   September 28


The Committee conducted an extensive review of standards from other states and nations, as well as research reports.  A full list is enclosed in the list of References.


B.        Charge to the Committee


The Committee's work has been guided by the Commissioner's Charge:


Charge to the Mathematics Standards Committee


The Committee will examine the existing Regents standards in mathematics, consider relevant research and other standards from the U.S. and other nations, and then propose modifications to the Regents mathematics standards to improve clarity, specificity, and functionality. The standards are to be challenging, and must represent a significant level of achievement in mathematics. The standards will "consist of a clear, well-defined set of skills, the mastery of which is demonstrable."  (Math A Panel Report, page 4) The Committee's recommendation will be consistent with the recommendations of the Independent Math A Panel.


Specifically we ask that committee members engage the following questions:


1.                  Is standard 3 of the NYS Mathematics, Science and Technology learning standards still sound?  Are the seven key ideas embedded in the mathematics standard all-inclusive?  Is there research/evidence to support modification of standard 3 and/or the seven key ideas? (See Attachment A: Seven Key Ideas.)


2.                  There is a presumption that there are too many performance indicators and that some should be consolidated and/or eliminated. Are the current performance indicators appropriate and sufficient for the developmental levels of elementary (grades K-4), intermediate (grades 5-8) and commencement (grades 9-12)?  Which performance indicators need to be added, revised, consolidated or eliminated at each level? 


3.                  What additional information needs to be provided to teachers to clarify the depth and breadth of understanding required of students for particular performance indicators?


4.                  What are the particular areas of study that should be taught in greater depth for greater understanding? Are there areas of study that should be eliminated?


5.                  In response to the NCLB requirements for grade-by-grade testing of mathematics in grades 3-8, the Standards Committee will develop content standards for grade six which will help inform the test development process for the new exams.

C.        Committee’s Response to its Charge



Charge Element

Committee’s Response

1.  Is standard 3 of the NYS Mathematics, Science and Technology learning standards still sound?  Are the seven key ideas embedded in the mathematics standard all-inclusive?  Is there research/evidence to support modification of standard 3 and/or the seven key ideas? (See Attachment A: Seven Key Ideas.)


The Committee’s recommendations include modifying the wording of Standard 3, and replacing the seven key ideas with five content strands and five process strands.  These recommendations were reached based upon a review of standards from other states and other nations, and also on extensive feedback from the field regarding the clarity of the wording of the current standards.

2.  There is a presumption that there are too many performance indicators and that some should be consolidated and/or eliminated. Are the current performance indicators appropriate and sufficient for the developmental levels of elementary (grades K-4), intermediate (grades 5-8) and commencement (grades 9-12)?  Which performance indicators need to be added, revised, consolidated or eliminated at each level?

The Committee is recommending revisions to the performance indicators for all grade levels, based on comparisons with other states and nations.


3.  What additional information needs to be provided to teachers to clarify the depth and breadth of understanding required of students for particular performance indicators?

The Committee’s recommendations address this question and focus on the need for grade-by-grade curriculum guidance, as well as the need for additional professional development.

4.  What are the particular areas of study that should be taught in greater depth for greater understanding? Are there areas of study that should be eliminated?

Based on its review of the mathematics standards in other states and nations, the Committee concluded that it is essential to revise the current program to enable students to understand concepts at a deeper level of understanding.  The recommended revisions reflect that conclusion.

5.  In response to the NCLB requirements for grade-by-grade testing of mathematics in grades 3-8, the Standards Committee will develop content standards for grade six which will help inform the test development process for the new exams.

The recommended revisions encompass all grades, including grade six.






A.        Guiding Principles


This Committee’s work was guided by a review of standards from a number of states in our nation, as well as standards and curricula from other nations of the world.  We were impressed by the focus of the mathematics programs in some other nations as compared with our current New York State standards.  Our standards include several topics each year, causing teachers and children to jump from topic to topic, with rarely enough time to cover material in any great depth.  We worked very hard to rewrite the performance indicators to provide teachers with sufficient time to focus on important areas of mathematics, so that children will be able to reach deep levels of understanding, and become mathematically proficient, which is so essential for future success.


The Mathematics Standards Committee based its work and recommendations upon a number of guiding principles:







The result of our work is a draft of a set of performance indicators in mathematics that the committee believes delineates the levels all students should attain.



B.        Time Constraints


The Committee has been concerned about the tight time constraints under which we have worked.  The development of standards for mathematics instruction at all grade levels, pre-K–12, is an enormous task.  Standards committees typically have two or three years to complete their work.  Our timeline, though, was "squeezed" by two events.  At the front end was the large failure rate on the June 2003 Math A Regents exam, which resulted in the appointment of the Math A Panel. The Panel presented its report in October 2003, at which time our formation was recommended.  At the back end is the federal “No Child Left Behind” law (NCLB), which requires that all school districts in the country test every child in mathematics in grades 3 – 8 beginning in the 2005-2006 school year.  For a test to be administered in March 2006, the test development work should have started around March 2004.  At that point, though, our group had just started its work. The Committee and the State Education Department were left with a difficult choice:  either take more time than we did to develop the new standards – which would have caused the new NCLB tests to be based on the old standards, or move as quickly as possible and work with the testing company to delay its timeline for test development. Our preference is the latter; the response from the field, and the Math A Panel report, dictate revisions to the current standards.



C.        Field Review of Our Work is Recommended


The Committee believes the standards we have developed, in addition to meeting all of the guiding principles, will be very helpful to teachers because of their simplicity and clarity.  However, from our first meeting, and to this day, we have felt that statewide review is essential before finalization of the document.  Such public review can only strengthen our work which, in turn, will strengthen mathematics education for New York State's children.



D.        The Importance of "The Power and Beauty of Mathematics"


The Committee anguished over whether to include wording in MST Standard 3 that would communicate the importance of children learning to appreciate the power and beauty of mathematics.  This should be the goal of every person who teaches mathematics, from the early childhood teacher to the high school calculus teacher.  Excitement in any classroom is contagious; and mathematics is fun and exciting to study.  Children should learn that mathematics is elegant and beautiful.  Helping children see this beauty is as important as helping children see the beauty of a daVinci painting or a Beethoven Symphony.


We considered including language in the standards establishing this expectation.  We were dissuaded because we were informed that any statement in a standard could be assessed, and children could be held accountable for meeting the standard.  We know of no way to assess the “power and beauty” of mathematics; and even if there is a way to assess it, we do not believe children should be denied a high school diploma for failing to appreciate this "power and beauty."  There was strong sentiment from several of our members that it is sad that the current national focus on assessment prevents us from including language stating this as an expectation for students.  Let there be no misunderstanding; we unanimously and strongly support the notion that an overarching goal of all mathematics instruction must be to help children appreciate the power, the beauty, and the elegance of mathematics, and we urge teachers to do all that they can to inspire their students to acquire this appreciation.






A.        The Standards



            1.         MST Standard 3


Discussion.  The charge to our Committee asked us to ". . . propose modifications to the Regents mathematics standards to improve clarity, specificity, and functionality."  As we reviewed the wording of MST Standard 3, which articulates the mission of mathematics education in our state, we felt that changes would improve clarify and functionality.  This leads to our first recommendation.


Recommendation 1.  The Committee recommends that MST Standard 3 be rewritten as follows:


Students will:


·        understand the concepts of, and become proficient with the skills of, mathematics

·        communicate and reason mathematically

·        become problem solvers by using appropriate tools and strategies


through the integrated study of number sense and operations, algebra, geometry, measurement, and statistics and probability.



            2.         Key Ideas


Discussion.  The current seven key ideas form the framework for the performance indicators.  As we reflected upon the current key ideas, we saw that they attempted to interweave content standards and process standards.  Our Committee felt it would be helpful to distinguish between the two types of standards to give clearer guidance to the field.  As we embarked upon this work, we concluded that the content standards would be most useful if they reflected the commonly understood branches of mathematics.  Regarding the process standards, we felt the NCTM (National Council of Teachers of Mathematics) statements were very clear, and we felt New York State should adopt these.  As we envision this, we see the two sets of standards being applied simultaneously, for example, while teachers work to help students attain mastery of the content of algebra, the five process strands (Problem solving, Reasoning and Proof, Communication, Connections and Representation) should be part of the instruction and of the assessment.   This leads to our second recommendation.



Recommendation 2.  The Committee recommends that the current seven key ideas be replaced with the following five content strands and five process strands:


Content Strands


Process Strands

o       Problem solving



            3.         Detailed Performance Indicators


Discussion.  Once MST Standard 3 and the ten strands (five content and five process) are determined, the work moves into great detail, too much detail for this report introduction.  The Committee spent enormous time developing the grade-by-grade detail which we believe will give clear direction to the field.  This work is included as Attachment B to this introduction.


Recommendation 3.  The Committee recommends that the standards document accompanying this report as Attachment B be adopted as a draft.  This document includes definitions of each content strand, definitions of each process strand, the division of each content strand into bands, and grade-by-grade performance indicators keyed to the strand.[1],[2]  We further recommend that this document be made available to the field for review and feedback before finalization.




            4.         Implementation Timeline for 3-8 and High School


Discussion.  The new standards and performance indicators will guide both the curriculum and the state assessments for mathematics in grades 3-8.  Under NCLB, these new assessments must first be administered in school year 2005-2006.  The committee believes that additional time would enable more reflection and review, and ultimately, in an improved product.  More time would also allow additional time for curriculum development, professional development and program transition, thus, Recommendation 4.


Recommendation 4.  The Committee recommends that the State Education Department consider requesting from the federal government a waiver for a one-year postponement of the NCLB math tests in grades 3-8.  If this is not possible, the Committee feels it is imperative that all accountability measures be adjusted or delayed so that there is ample phase in time for these revised standards.


Discussion.  The Committee further believes that a phase-in approach for the high school program is important.  Because success in the high school courses will depend on the skills and knowledge learned in grades 3–8, we believe the high school changes should not be implemented until a year after the 3-8 program.  Also, the Committee believes, because Math B depends on Math A, and because the changes recommended herein will change these programs, the implementation of the new Math B should be made the year after the implementation of the new Math A.  Recommendation 5 addresses this, and includes a table illustrating the timelines with or without the waiver recommended in Recommendation 4.


Recommendation 5.  The Committee recommends that the high school program be phased in over a three year period, following the implementation of the 3-8 program by one year.



Implementation Year


One Year Waiver

Implementation Year


One-Year Waiver

Grades 3-8



Math A



Math B





B.        High School Mathematics Program, and Graduation Requirements



            1.         Graduation Requirements and Regents Examinations


Discussion.  The Committee believes that competency in mathematics is critically important for success in our society. The Committee, therefore, endorses current Regents policy establishing a graduation requirement of requiring students to take three units of mathematics and to pass one Regents mathematics examination.


The Committee also applauds the Board for approving in October, 2003 Recommendation 2 of the Math A Panel to restructure Math A:


Recommendation 2:   The [Math A] standards and curricula should be structured so that the typical student will take the Math A exam after one year of high school mathematics.


This Committee endeavored to establish standards for Math A to meet this objective, i.e., that it be a one-year course for the typical student.  If Math A is a one year course for the typical student, and if students are required to take three units of mathematics in high school, then it follows that Math B will take that typical student two years to learn.  This has some of the same inherent problems found by the Math A Panel, e.g., many students will find that the first half of Math B is taught by one teacher, and the second half by another. Without some measure of the progress at the end of the first half of Math B, the teacher of the second half cannot be sure students are continuing with the requisite skills and knowledge.  The Committee believes it is very important that there be external validation of the first half of Math B in the form of an additional Regents exam.  Without an additional state exam at the end of the first half of Math B, the typical student will be tested in grades 3, 4, 5, 6, 7, 8, 9, and 11 -- but not grade 10.

Recommendation 6. The Committee recommends that an additional Regents exam be created to test students at the end of the first half of Math B.



            2.         Course Content and Labels


Discussion.   The Standards Committee explored a number of options for the framework for the Math A performance indicators.  The challenge is twofold.  First, Math A must provide a strong foundation for those students continuing to Math B, and also for those students who choose other, locally developed mathematics courses.  After a great deal of deliberation, the Committee concluded that changes to Math A are warranted, for two main reasons.  First, Math A currently includes content from several branches of mathematics, which we have heard over and over again from teachers requires them to jump from topic to topic.  Second, the strongest international mathematics programs give teachers enough time to spend on a topic to enable deep learning.  Therefore, we decided that it was important to redesign Math A to focus primarily on one branch of mathematics.  The committee could not find research to support one branch of mathematics over another; however, the years of experience of the group, including those who practice mathematics in their professions, led us to the unanimous conclusion that the focus of Math A should be on algebra.  However, recognizing that every person in our society needs to know more than algebra, the Committee added expectations that go beyond algebra.  Additionally, in order to ensure that the body of knowledge students must know to receive a diploma is robust, the Committee also moved several algebra standards from the current Math A into earlier grades so that, when students pass the Math A Regents exam, they will be demonstrating substantial knowledge and skills in mathematics.  In short, the Committee believes that the proposed Math A is robust, and much more focused than the current Math A.


The Committee then wrestled with Math B, which the typical student will take over a two-year period.  The Committee felt substantial changes to Math B were necessary, for example, the inclusion of more geometry to ensure a deep understanding of geometric relationships.


As our work progressed, the Committee recognized that the new focus of Math A and Math B provided an opportunity to address a long standing problem in New York State, which is that the labels we have assigned to our mathematics programs do not describe the mathematics being taught and learned.  In science, we have Earth Science, Living Environment, Chemistry and Physics.  In social studies, we have Global Studies and U.S. History and Government.  These labels are descriptive.  In mathematics, though, we have Math A and Math B (or, previously, Course I, Course II and Course III or, before that, Math 9, Math 10, and Math 11).  Such non-descriptive labels create enormous communications problems in a wide variety of situations, from outside observers trying to understand what mathematics we are expecting of our students in New York, to students moving in or out of the state trying to have a smooth math program transition, to the NCAA evaluating New York's students.  Our Committee was charged to ". . . propose modifications to the Regents mathematics standards to improve clarity, specificity, and functionality."  We believe a major step toward that end would be to replace New York State's non-descriptive, non-standard high school math course labels with ones that are easily understood by everyone, from students, to college admissions officers, to mathematics educators in other states and nations.  In line with this thinking, as we developed the proposed standards, we saw that the expectations we are recommending for Math A could appropriately be labeled "Integrated Algebra," as the course focuses on algebra knowledge, while connecting algebra with other branches of mathematics and while including the process strands.  Similarly, the expectations we are recommending for Math B could be appropriately labeled "Integrated Geometry, Integrated Algebra II and Trigonometry."


Further, as we discussed this, and as our work progressed, we connected two thoughts, i.e., (a) Math B should be conceptualized as a two-year course for the typical student and (b) our course labels should be clear.  As Math B consists of a set of standards in the areas of Integrated Geometry, Integrated Algebra II and Trigonometry, these can easily be split into two sets of sequential standards, the first being Integrated Geometry and the second being Integrated Algebra II and Trigonometry.  Consistent with the Math A Panel recommendations, by splitting Math B into these two broad sets of topics, we believe the typical student can master each of the two halves of the standards in a course with a one-year duration.  Thus,  we believe the call for clearer standards that will be more easily understood by teachers, parents and students logically extends to the recommendation that Math A and Math B be replaced by the simple and easily understood terms, "Integrated Algebra," "Integrated Geometry" and "Integrated Algebra II and Trigonometry."  This discussion leads to the following recommendation.


Recommendation 7.  The Committee recommends that, consistent with the performance indicators presented in Attachment B, the labels for high school mathematics be changed as follows:




            3.         Additional High School Program Issues


Discussion.  As the Committee envisions the high school program that will result from this set of recommendations, there will be three courses, each building on the knowledge and the skills of the earlier ones.  We believe it is important that local districts take steps to ensure that students have sufficient mastery of each before moving to the next.


Recommendation 8.  The Committee recommends that each local district establish criteria based on the course grades and Regents examination scores for students to continue to the next course in the Math A – Math B (Integrated Algebra, Integrated Geometry, Integrated Algebra II and Trigonometry) sequence.


Discussion.  Some students may struggle with Math A (Integrated Algebra) and may take three years of coursework to pass the Regents examination.  The Committee believes all students should take mathematics beyond Math A and reached the conclusion that the number of units of credit that should be granted for Math A should be limited to two.  Thus, every student will take at least one year of mathematics beyond Math A.


Recommendation 9.  The Committee recommends that the amount of course credit that can be granted by local districts for Math A (Integrated Algebra) should be limited to two units.


Discussion.  It is important that students who choose alternatives to Math B, after having passed the Math A Regents exam, receive high quality alternatives to Math B.


Recommendation 10.  The Committee recommends that school districts, teachers administrators and guidance counselors emphasize the importance of mathematics in our modern society, and that they encourage children to take the three core courses named above for a Regents Diploma with Advanced Designation.  Recognizing that not all students will follow this path, however, the Committee recommends that, for those students who have passed one mathematics Regents examination and who choose not to take the three year Regents Diploma with Advanced Designation sequence, the additional required coursework should be locally developed, based on frameworks established by the State Education Department.  This Committee would have worked on this but time did not permit.  We recommend that SED establish a group to develop this framework.  This Committee sees this framework as including a majority of these topics:  ratios and proportions, statistics, data analysis, informal geometry, number theory, financial applications (financial management, tax structures, etc.), and integration of technology.



C.        Guidance to Classroom Teachers on the Standards


Recommendation 11.  This Committee believes it is essential that grade-by-grade curriculum guides be developed as suggested models for the field and that there be alignment among the standards and performance indicators, the curricula, and the assessments. (Please note that this has the same intent as the Math A Panel's Recommendation 1C.)


These curriculum guides should include a list of topics to be taught, sample lessons, pedagogical suggestions, and a suggested time frame for each topic.  Because of the months of work this Committee has dedicated to the development of the standards and performance indicators, the Committee believes its members should be connected with the development of a suggested curriculum, to ensure that the curriculum is consistent with the standards and performance indicators.



D.        Technology and Mathematics Instruction


The Mathematics Standards Committee believes that the use of technology is an important and valuable tool in the mathematics classroom and we endorse the spirit of the technology statement included in the original Mathematics Resource Guide with Core Curriculum.


All mathematical skills require knowledge of the fundamentals.  When appropriately incorporated into classroom instruction, technology can be an effective classroom tool and can enhance learning.  We define technology as electronic equipment that can be used to advance teaching and learning such as, but not limited to, four-function calculators, scientific calculators, graphing calculators, computer software (e.g. spreadsheets, statistical programs, dynamic geometry software, etc.), digital cameras, GPS, video, and multimedia, etc.[3]


Technology is a powerful student motivator; it helps students visualize concepts and ideas. Technology helps teachers engage students with different learning styles understand mathematical concepts and ideas. Through the use of technology, students are able to form personal references to mathematical concepts.


The National Center for Educational Statistics, NCES, a part of the U.S. Department of Education, reports that by the fall of 2000, 98% of U.S. public schools had access to the internet.[4] Technology is here to stay and is continually changing. Technology can erase equity issues if all students have access to it.  To assure that all students have equitable access to technology, there must be adequate state funding.


The use of a calculator is essential. Research has supported the use of calculators and other technologies, in learning mathematics at an early age and beyond.[5]  Findings from TIMSS and other research show a strong relationship between calculator use and achievement at the secondary level.[6]


Technology Recommendations


Recommendation 12 A.    The Committee recommends the continued use of four function calculators for grades K-4 as an instructional tool.  The Committee recommends that calculators not be used on the grade three and four assessments


Recommendation 12 B.     The Committee recommends the use of scientific calculators in grades 5-8. Use of calculators on the 5-8 assessments should be similar to the format that is currently used on the Grade 8 assessment, i.e., no calculator use permitted on Part I, and calculator use permitted on extended response items, with the understanding that all students will be given access to a scientific calculator.: On part I, no calculator use permitted and on extended response parts, calculator use permitted, with the understanding that all students will be given  access to a scientific calculator.


Recommendation 12 C.    The Committee recommends the exploratory use of graphing calculators in grades 7-8.


Recommendation 12 D.    The Committee recommends that graphing calculators be used on a regular basis in mathematics instruction in grades 9-12, recommends that their use be required on the Regents exams, and recommends that all students be provided access to a graphing calculator.


E.        A Capacity Issue – Adequately Prepared Teachers


Recommendation 13.  The success of these standards (or, for that matter, any educational standards) rests squarely on the shoulders of classroom teachers. It is imperative that the teachers responsible for helping children reach these standards be highly knowledgeable about mathematics and pedagogy.  The Committee strongly endorses Finding 9 and the pertinent recommendations made by the Independent Math A Panel, as stated in the Executive Summary (pp. 6, 7):


Finding 9:  The mathematical background of teachers delivering math instruction varies widely; yet, raising almost three million children to higher levels of math achievement will be impossible without highly skilled teachers.


Recommendation 9A:   SED and higher education need to continue and to strengthen their partnerships to ensure strong teacher education programs, both pre-service and in-service.

Recommendation 9B:   The certification requirements for elementary teachers and special education teachers should include a minimum of nine credits of college level mathematics (see Recommendation 9C), and three credits of teaching techniques in mathematics.

Recommendation 9C:   Mathematics courses required for certification, both for mathematics teachers and elementary and special education teachers, should be specific not only in terms of number of credits required to be taken, but also in terms of coursework required to be taken, e.g., calculus, number theory, algebraic structures, probability and statistics, etc.

Recommendation 9D:   The Panel believes that, for any teacher responsible for teaching mathematics at any level, the 175-hour professional development requirement should include specific mathematics requirements.  The Panel's thinking is that;

·             teachers who teach mathematics exclusively should be required to take 100 of the 175 hours in the area of mathematics;

·             secondary teachers who are certified in, and who teach in, more than one subject area, should be required to take 50 of the 175 hours in the area of mathematics;

·             teachers who teach mathematics as part of a broad set of teaching responsibilities, e.g., elementary teachers and special education teachers, should be required to take 30 of the 175 hours in the area of mathematics.


Additionally, the range of possible courses that would satisfy these requirements should be clearly specified.


This Committee recommends that a statewide Committee for the Preparation of Teachers of Mathematics be established to respond to Recommendations 9B and 9C.  The Committee also recommends that the Panel's Recommendations 9A and 9D be implemented without delay to assure that all of our children have teachers who have the knowledge and skills needed to help them reach these standards.




The Committee believes it is the performance indicators to which teachers will refer for guidance in their daily work and, therefore, we felt it might be helpful to review the process for developing these, and to highlight the major changes from the current program.


The new performance standards were based on the new model of five content strands and five process strands, which replaces the current model of seven key ideas.  As part of this effort, the Committee reviewed all of the current performance indicators.  The current standards document includes 289 broadly worded performance indicators, with each having one or more "includes" bullets as more specific examples.  There are 876 such "includes" bullets.  This Committee saw its work as developing specifically worded performance indicators, more similar to the "includes" bullets in the current document.  For the purpose of this discussion, therefore, the term "performance indicator," when applied to the current document, will be applied to the more specifically worded "includes" bullets.  Some of the current 867 performance indicators remained at the current grade level, either with the original wording, or with revisions to improve clarity.  Some were moved from one grade to another or replaced by others.  In total, our Committee wrote 850 performance indicators.  Unlike the current performance indicators,  which are in grade level bands, e.g., Prekindergarten-K, 1-2, 3-4, etc.,  our proposed performance indicators are presented grade by grade, to provide clarity and to increase functionality.


Our Committee developed these 850 performance indicators in three ways, first by writing them in three grade level groups (Prekindergarten-4, 5-8, 9-12), then by breaking into five groups defined by the five content strands to ensure vertical alignment, then by reviewing them before the whole Committee to ensure Prekindergarten-12 consistency.  Each of the current performance indicators was reviewed to ensure that no skill was inadvertently omitted in the proposed list.  The frequent review by grade level and by strand aligned the effort to produce performance indicators that the Committee believes are clear and specific, and which we believe will be functional for the field.


Given that our proposal shifts the 876 current performance indicators to 850 new performance indicators, it would not be possible to detail all of the changes from the current to the proposed in this summary document.  There are some shifts that should be highlighted.  First, with regard to Prekindergarten-4, while there are shifts of performance indicators, they are minor in nature, with a topic perhaps moving a grade earlier or later.  The emphasis in these grades is on building a strong foundation in numeracy.  At the middle school and high school, to strengthen the program, some significant shifts occurred.  First, some algebra topics have been moved into earlier grades.  For example, addition and subtraction of polynomials, factoring trinomials, and basic inequalities have been moved from Math A to 8th grade.  Similarly, some work with equations, e.g., solving one and two step equations, has been moved from the current grades 7‑8 to grades 5 and 6.  (While some in the field have questioned the ability of children to challenge some of these areas at these new grade levels, the educators on this Committee, many of whom work with children each and every day, are confident that, with adequate curriculum and staff development, this will be successful.  Indeed, many of us know public schools that are doing this successfully right now.)


At the high school, to provide focus for each of the high school courses, the Committee moved several topics from grade to grade.  Currently, Math A and Math B include a mix of many topics.  Our proposal, as noted above, is to focus the Math A – Math B program on three areas:  Algebra, Geometry, and Algebra II and Trigonometry.  Providing this focus necessitated moving current performance indicators from one grade to another, writing new ones, etc.  One shift was the movement of several performance indicators to the course consistent with the name (e.g., geometry indicators from Math A or Math B to Integrated Geometry).  Another shift was a deliberate effort to include statistics and probability with Integrated Algebra and Integrated Algebra II, but not in Integrated Geometry.  The purpose is to allow sufficient time in that course for geometric representations of algebraic expressions, equations or inequalities, an area which we felt was important and which should be connected closely with geometry.  Some specific examples of these connections include, "Write the equation of a vertical line" and "Use the graph of a parabola to find the solution set of a quadratic inequality with one variable."  Additionally, the proposed performance indicators at the high school level includes more focus on geometry in general, as the Committee believes geometry skills are valuable for our students.


These are some of the major changes reflected in the proposed performance indicators.  The curriculum guidance forthcoming from SED will provide additional information.

V.             CONCLUSION


This Mathematics Standards Committee appreciates the efforts of the State Education Department to reach out to the field to assist with the development of standards and performance indicators for children.  This work has been an enormously enriching experience for all of us.


We believe we have met the charge established by the Commissioner and that, if our recommendations are adopted, our state's mathematics expectations will be both clear to all, and robust.


We believe it is important to emphasize that the problems identified by the Math A Panel were systemic in nature.  It is imperative that all facets of the system be synchronized.   These facets include measurable standards with high expectations for student performance, alignment of suggested grade-by-grade curricula and assessments, and professional development to the standards.  These must all be in place.  Once they are in place, we believe New York State's children will be receiving the best mathematics education anywhere.  We hope our work has advanced this goal.



Academic Content Standards K-12 Mathematics. (2001). Ohio Department of Education.

Academic Standards for Mathematics. (2002). Pennsylvania Department of Education.

Achieve, The Education Trust and the Thomas B. Fordham Foundation.  (2004). The American Diploma Project. 

Braden, L., Finn, C., Lerner, L., Munroe, S., Petrilli, M., Raimi, R., Saxe, D., Smith, T., Stotsky. (2000).  The State of State Standards 2000.  The Thomas B. Fordham Foundation.

Burrill, G., Allison, J., Breaux, G., Kastberg, S., Leatham, K., & Sanchez, W. (2002). Handheld graphing technology at the secondary level: Research findings and implications for classroom practice. Dallas, TX: Texas Instruments.

Clements, D. & Sarama, J.  (2004). Engaging Young Children in Mathematics: Standards for Early Childhood Mathematics Education. Lawrence Erlbaum Associates, Publishers.

Clements, D., Sarama, J. & Lee, J.  Proposal Phase II - Scaling Up the Building Blocks Pre-K Mathematics Curriculum: Teaching for Understanding with Trajectories and Technologies.

Cohen, M., Finn, C. & Haycock, K. (2004). Creating a High School Diploma That Counts.  Education Week.

Content Standards for California – Mathematics. (2001). California Department of Education.

Grade Level Expectations for the Sunshine State Standards – Mathematics. (1996). Florida Department of Education.

Griffin, S. (2004). Teaching Number Sense. Educational Leadership, Association for Supervision and Curriculum Development.  pp. 39-42.

Harskamp, E., Suhre, C., & Van Streun, A. (2000). The graphics calculator and students’ solution strategies. Mathematics Education Research Journal, 12(1), 37-52.

Hollar, J.C., & Norwood, K. (1999). The effects of a graphing-approach intermediate algebra curriculum on students’ understanding of function. Journal of Research in Mathematics Education, 30(2), 220-226.

Illinois Learning Standards for Mathematics. (1997). Illinois State Board of Education.

Indiana’s Academic Standards. (2003).  Indiana Department of Education.

International Baccalaureate Organization.  (1997).  Mathematical Studies Standard Level: September 1997.  International Baccalaureate.

Kilpatrick, J., Swafford, J.,  & Findell, B, editors (2001).  Adding it Up: Helping Children Learn Mathematics. National Academy Press.

Larson, M.  (2003).  Improving the Mathematics Achievement of All Students: What We Know from the Research. Issue in Brief.  National Association of State Boards of Education.

Lee Peng Yee and Fan Lianghuo.  (2002).   The Development of Singapore Mathematics Curriculum:  Understanding the Changes in Syllabus, Textbooks and Approaches.  National Institute of Education, Singapore.

Loveless, T. & Coughlan, J. (2004).  The Arithmetic Gap. Educational Leadership, Association for Supervision and Curriculum Development.  pp. 55-58.

Marzano, R.J.  (2003).  What Works in Schools: Translating Research into Action.  Association for Supervision and Curriculum Development.

Massachusetts Mathematics Curriculum Framework. (2000). Massachusetts Department of Education.

Michigan Curriculum Framework. (1996) Michigan Department of Education

Mid-continent Research for Education and Learning. (2001). A Technical Guide for Revising or Developing Standards and Benchmarks. Regional Educational Laboratory. Office of Educational Research and Improvement.

Minnesota K-12 Mathematics Framework. (1998). SciMath. Minnesota Department of Children, Families & Learning.

Nagasaki, E., Sawads, T., & Senuma, H. (1990).  Mathematics Program in Japan.  Japan Society of Mathematical Education.,00.shtm

National Assessment of Educational Progress. (2003). 2005 NAEP Mathematics Assessment and Item Specifications. The National Assessment Governing Board. 

National Center for Education Statistics (2002), Internet Access in U.S. Public Schools and Classrooms: 1994 – 2001.

National Center for Educational Statistics. (2001). Highlights from the TIMSS 1999 Video Study of Eighth-Grade Mathematics Teaching. U.S. Department of Education.  Institute of Education Sciences.

National Center for Education Statistics, National Assessment of Education Progress (NAEP), 2000 Mathematics Assessment.

National Center for Educational Statistics. (2003). The National Assessment of Educational Progress - Mathematics – The Nation’s Report Card. U.S. Department of Education.  Institute of Education Sciences. 

National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics.

National Research Council. (2001). Adding It Up – Helping Children Learn Mathematics. National Academy Press.

National Research Council.  (1989).  EVERYBODY COUNTS: A Report to the National on the Future of Mathematics Education.

National Research Council. (2002). Helping Children Learn Mathematics. National Academy Press.

New Jersey Core Curriculum Content Standards for Mathematics. (2002). New Jersey Department of Education

New York State Education Department. Mathematics Resource Guide with Core Curriculum (1998),The University of the State of New York.

New Standards – New York City Performance Standards  (First Edition). (1996). New York City Board of Education

New York State Education Department. (2003). A Report to the Governor and the Legislature on the Educational Status of the State’s Schools:  Submitted July 2003.  The University of the State of New York.

Olson, L. (2004). States Must Beef Up Diploma Demands, Study Maintains.  Education Week.

Ruthven, K. (1990). The Influence of Graphic Calculator Use on Translation from Graphic to Symbolic Forms. Educational Studies in Mathematics, 21, 431-450.

Schmidt, W., McKnight, C., & Raizen, S. (1997). A Splintered Vision: An Investigation of U.S. Science and Mathematics Education.  Kluwer Academic Publishers.

Schmidt, W. (2004). A Vision for Mathematics.  Educational Leadership, Association for Supervision and Curriculum Development.  pp. 6-11.

Solomon, P. (2001).  The Math We Need to “Know” and “Do.”  Corwin Press.

Southern Regional Education Board. (2002). Getting Students Ready for Algebra I: What Middle Grades Students Need to Know and Be Able to Do.

Standard Course of Study and Grade Level Competencies K-12. (Revised 2003). State Board of Education. Public Schools of North Carolina. Department of Instruction.

Texas Essential Knowledge and Skills (TEKS) Learning Standards for Texas Children. (1998). Texas Education Agency.

The Connecticut Framework K-12 Curricular Goals and Standards. (1998). Connecticut State Department of Education. Division of Teaching and Learning.

Thomas B. Fordham Foundation.  (2004). Grading the Systems: The guide to state standards, tests, and accountability policies.

Thompson, D. R., & Senk, S. L. (2001). The effects of curriculum on achievement in second-year algebra: The example of the University of Chicago School Mathematics Project. Journal for Research in Mathematics Education, 32(1), 58-84.

Wilson, Suzanne. (2003). California Dreaming: Reforming Mathematics Education. Yale University Press.

Attachment A


Current Learning Standard and Key ideas


Learning Standards for Mathematics, Science and Technology


Standard 3:  Students will understand mathematics and become mathematically confident by communicating and reasoning mathematically, by applying mathematics in real-world settings, and by solving problems through the integrated study of number systems, geometry, algebra, data analysis, probability and trigonometry.


Seven Key Ideas

  1. Mathematical Reasoning

Students use mathematical reasoning to analyze mathematical situations, make conjectures, gather evidence and construct an argument.

  1. Number and Numeration

Students use number sense and numeration to develop an understanding of the multiple uses of numbers in the real world, the use of numbers to communicate mathematically, and the use of numbers in the development of mathematical ideas.

  1. Operations

Students use mathematical operations and relationships among them to understand mathematics.

  1. Modeling/Multiple Representation

Students use mathematical modeling/multiple representation to provide a means of presenting, interpreting, communicating, and connecting mathematical information, and relationships.

  1. Measurement

Students use measurement in both metric and English measure to provide a major link between the abstractions of mathematics.

  1. Uncertainty

Students use ideas of uncertainty to illustrate that mathematics involves more than exactness when dealing with everyday situations.

  1. Patterns/Functions

Students use patterns and functions to develop mathematical power, appreciate the true beauty of mathematics and construct generalizations that describe patterns simply and efficiently.



Learning Standard – an established level or degree of quantity, value or quality.  The NYS Learning Standards are defined as the knowledge, skills and understandings that individuals can and do habitually demonstrate over time as a consequence of instruction and experience.


Key Idea – major domains (skills, knowledge or ideas) that define fields of study or areas of learning.  The NYSED key ideas define their respective learning standards in specific content areas (e.g., dance, music, theatre, visual arts) or fundamental skills (e.g., reading, writing, listening, speaking).


Performance Indicator – a description of student achievement expectations of the developmental levels of elementary (grades K-4), intermediate (grades 5-8) and commencement (grades 9-12).


[1] The Committee wishes to make it clear that the order of presentation of the performance indicators in this draft should not necessarily be read as the order in which topics should be presented during the school year.  It is our expectation that the mathematics curriculum committee will develop a curriculum that appropriately sequences each of the performance indicators during the school year.


[2] Recognizing that the NCLB testing is planned to be administered to all children in grades 3-8 in March (beginning in 2006), the Committee has endeavored to identify in this draft those performance indicators which it sees as important to be attained before March, and those which it perceives can be demonstrated after March.  The purpose of dividing the performance indicators into pre-March and post-March is to provide direction, so that both test developers and classroom teachers will know which performance indicators are most likely to be assessed on the March tests.


[3] See SED Mathematics Resource Guide with Core Curriculum (1998) p. 8.


[4] See National Center for Education Statistics Report (2002), Internet Access in U.S. Public Schools and Classrooms: 1994-2001.


[5] Clements and Sarama (2004), Recommendation 9. See also NCTM (2000) and National Research Council (2001) Adding it Up: Helping Children learn Mathematics, Chapter 9: Teaching For Mathematical Proficiency.


[6] Burrill et al. (2002): Harskamp, et al. (2000): Hollar and Norwood. (1999): Ruthven. (1990): Thompson and Senk. (2001).