I.
INTRODUCTION
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.
|
II.
IMPORTANT POINTS RELATIVE TO OUR WORK
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
performance indicators must be presented on a grade-by-grade basis, rather
than in bands of grades, as was previously the case. The purpose is to provide vitally
important information to classroom teachers across New York State, and to
clarify the expectations for parents and students for each grade.
- The
number of performance indicators in each given year should be developed with
the goal of enabling teachers to focus on important areas of mathematics, so
that they can be sure the children have reached deep levels of
understanding.
- The
language should be clear for teachers, parents, students and test developers,
both in terms of the expectations, and in terms of the depth of understanding
of each expectation. If everyone
understands the expectations, then everyone will be working toward the same
goal.
- The
performance indicators need to be sequenced appropriately across the grades,
so that knowledge and skills grow from year to year.
- The
performance indicators must be developmentally appropriate at each grade
level.
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.
III.
RECOMMENDATIONS
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
- Number Sense and
Operations
- Algebra
- Geometry
- Measurement
- Statistics and
Probability
Process
Strands
o
Problem
solving
- Reasoning and Proof
- Communication
- Connections
- Representation
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., 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
With
One Year Waiver |
Implementation Year
Without
One-Year Waiver |
|
Grades 3-8 |
2006-2007 |
2005-2006 |
|
Math A |
2007-2008 |
2006-2007 |
|
Math B |
2008-2009 |
2007-2008 |
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:
- From
Math A to "Integrated Algebra"
- From
Math B to "Integrated Geometry" for the first half and "Integrated Algebra
II and Trigonometry" for the second half.
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.
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.
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. Findings from TIMSS and other research
show a strong relationship between calculator use and achievement at the
secondary level.
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.
IV.
SUMMARY OF CHANGES TO PERFORMANCE
INDICATORS
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.
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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
- Mathematical
Reasoning
Students use mathematical reasoning to
analyze mathematical situations, make conjectures, gather evidence and construct
an argument.
- 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.
- Operations
Students use mathematical operations and
relationships among them to understand mathematics.
- Modeling/Multiple
Representation
Students
use mathematical modeling/multiple representation to provide a means of
presenting, interpreting, communicating, and connecting mathematical
information, and relationships.
- Measurement
Students use measurement in both metric
and English measure to provide a major link between the abstractions of
mathematics.
- Uncertainty
Students use ideas of uncertainty to
illustrate that mathematics involves more than exactness when dealing with
everyday situations.
- 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.
Definitions:
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).
