```Washington’s Math Standards
David Klein
Professor of Mathematics
California State University, Northridge
Why do standards matter?
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goal posts for teaching and learning
determine the content and emphasis of tests
influence the selection of textbooks
form the core of teacher education programs
The State of State Math
Standards 2005
Fordham Foundation
Co-authors of the Fordham Foundation Report:
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Bastiaan Braams, Emory University
Thomas Parker, Michigan State University
William Quirk, Ph.D. in Mathematics
Wilfried Schmid, Harvard University
W. Stephen Wilson, Johns Hopkins University
What’s Wrong with
Washington's Standards?
Excessive use of calculators, standard algorithms
missing, poor development of fractions and
decimals, weak algebra standards (little more than
linear equations), very little geometric reasoning
and proofs, weak problem solving standards, too
many standards unrelated to math
Standards with little relationship to math:
• Determine the target heart zone for participation in aerobic
activities.
• Determine adjustments needed to achieve a healthy level of
fitness.
• Explain or show how height and weight are different.
• Explain or show how clocks measure the passage of time.
• Explain how money is used to describe the value of purchased
items.
• Explain why formulas are used to find area and/or perimeter.
• Explain a series of transformations in art, architecture, or
nature.
• Recognize the contributions of a variety of people to the
development of mathematics (e.g. research the concept of the
golden ratio).
Focus: talking about solving problems, rather than actually
solving problems.
Long lists of vague, generic tasks: "Gather and organize the
necessary information or data from the problem," "Use
strategies to solve problems," "Describe and compare strategies
and tools used," "Generate questions that could be answered
using informational text".
Misleading: one does not learn how to solve problems by
following these outlines.
Useless: little indication of which types of problems students
are expected to solve.
A classroom is presenting a play and everyone has invited
two guests. Enough chairs are needed to seat all the
guests. There are some chairs in the classroom.
believe that the women will run as fast as the men” in the
Olympics.
Given: a list of running times of men and women, for an
unspecified distance for several years of Olympic games.
No further information.
Calculators
“Technology should be available and used throughout the K–12
mathematics curriculum. In the early years, students can use
basic calculators to examine and create patterns of numbers.”
Solve problems involving addition and subtraction with
two or three digit numbers using a calculator and
explaining procedures used.
Fractions
Introduced for the first time in 4th grade:
Explain how fractions (denominators of 2, 3, 4, 6, and 8) represent
information across the curriculum (e.g., interpreting circle graphs,
fraction of states that border an ocean).
Fifth graders use calculators to multiply decimal numbers before
they learn meaning of fraction multiplication.
What does it mean to multiply fractions, in particular, decimals?
The answer comes a year later. This is rote use of technology
without mathematical reasoning.
Explain the meaning of multiplying and dividing nonnegative fractions and decimals using words or visual or
physical models (e.g., sharing a restaurant bill, cutting a
board into equal-sized pieces, drawing a picture of an
equation or situation).
Division of fractions is often incorrectly defined as
repeated subtraction. E.g. “cutting a board into equal
sized pieces.” Widely used CMP 6th grade textbook treats
fraction multiplication and division poorly, but is
considered to be aligned to Washington's standards
Patterns
“What is Mathematics? - Mathematics is a language and science of
patterns.”
“As a language of patterns, mathematics is a means for describing the world
in which we live. In its symbols and vocabulary, the language of mathematics
is a universal means of communication about relationships and patterns.”
“As a science of patterns, mathematics is a mode of inquiry that reveals
fundamental understandings about order in our world. This mode of inquiry
relies on logic and employs observation, simulation, and experimentation
as means of challenging and extending our current understanding.”
-- Office of the Superintendent of Public Instruction
www.k12.wa.us/curriculumInstruct/mathematics/default.aspx
• Recognize or extend patterns and sequences using
operations that alternate between terms.
• Create, explain, or extend number patterns
involving two related sets of numbers and two
multiplication, or division.
• Use rules for generating number patterns (e.g.,
Fibonacci sequence, bouncing ball) to model reallife situations.
• Use technology to generate patterns based on two
arithmetic operations. Supply missing elements in
a pattern based on two operations.
• Select or create a pattern that is equivalent to a given
pattern.
• Describe the rule for a pattern with combinations of two
arithmetic operations in the rule.
• Represent a situation with a rule involving a single
operation (e.g., presidential elections occur every four
years; when will the next three elections occur after a
given year).
• Create a pattern involving two operations using a given
rule.
• Identify patterns involving combinations of operations in
the rule, including exponents (e.g., 2, 5, 11, 23).*
*Note: 3 x 2n – 1 and 1/2 (4 + 5n + n3) both give these values starting with n = 0
of number tiles. She used
a rule to create the pattern
in the number tiles.
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Extend the pattern to
complete the next row of
the triangle.
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Describe the rule you
used to extend the pattern.
Why are most state standards, including
The National Council of Teachers of Mathematics (NCTM)
has immense influence on state education departments and
K-12 mathematics education in general.
Most state standards adhere closely to guidelines published
by the NCTM:
• An Agenda for Action (1980),
• Curriculum and Evaluation Standards for School
Mathematics (1989)
• Principles and Standards for School Mathematics
(2000).
An Agenda for Action
• problem solving should be the focus of school
mathematics.
• “difficulty with paper-and-pencil computation should
not interfere with the learning of problem-solving
strategies.”
throughout their school mathematics program”
• “decreased emphasis on...performing paper and pencil
calculations with numbers of more than two digits.”
• de-emphasis of calculus
1989 NCTM Standards
• “The new technology not only has made calculations and graphing easier, it
has changed the very nature of mathematics . . .”
• “appropriate calculators should be available to all students at all times”
• More emphasis: “collection and organization of data,” “pattern recognition
and description,” and “use of manipulative materials”
• Less emphasis (K-4):“long division,” “paper and pencil fraction
computation,” “rote practice,” “rote memorization of rules,” and “teaching
by telling”
• Less emphasis (5-8):“manipulating symbols,” “memorizing rules and
algorithms,” “practicing tedious paper-and-pencil computations,” “finding
2000 NCTM Standards decreased the extreme
rhetoric but continued to promote the same
themes: calculators, patterns, manipulatives,
estimation over exact calculation and standard
algorithms and coherent development of math.
But,
2006 NCTM Focal Points are a step in the
right direction.
NCTM-Aligned Books
Mathematicians on Textbooks
November 1999: more than 200 university mathematicians added
their names to an open letter to the U.S. Education Secretary calling
upon him to withdraw recommendations for NCTM aligned
textbooks, including Connected Math, Core-Plus, and IMP.
The list of signatories included seven Nobel laureates and winners of
the Fields Medal, as well as math department chairs of many of the
top universities in the U.S., and several state and national education
leaders. Seven of the signers of this letter now serve on the
National Mathematics Panel.
NCTM President Johnny Lott in 2004 posted a denunciation of the
open letter on the NCTM website, under the title, “Calling Out” the
Stalkers of Mathematics Education:
Consider people who use half-truths, fear, and innuendo to
control public opinion about mathematics education. As an
example, look at Web sites that continue to use a public letter
written in 1999 to then Secretary of Education Richard Riley by a
group of mathematicians and scientists defaming reform
mathematics curricula developed with National Science
Foundation grants. . . A small group continues to use the letter in
an attempt to thwart changes to mathematics curricula.
Suggested problems for the students:
1/5 + 1/4 =
3/8 + 3/4 =
5/6 – 1/3 =
3 – 11/4 =
“These are the most difficult addition/subtraction problems for
fractions I could find in the TERC 5th grade curriculum (which is
described as ‘also suitable for 6th grade’)”--Wilfried Schmid, Dept.
of Mathematics, Harvard University
Math Wars: Points of Emphasis
• Mathematical Content
• Skills-based
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cohesion, clarity
proof
whole class instruction
teacher centered
model: university
parents, mathematicians,
• Pedagogy
• learn skills as needed for
“real world” problems
• discovery learning
• learning styles
• small group learning
• child centered
• model: kindergarten
• colleges of education,
Mathematical Competence
Mathematical ignorance of district and state K-12
math leadership is the single greatest barrier to
effective education.
Most math curriculum experts are mathematically
weak. They make bad decisions, not only because
they blindly follow the NCTM standards, but also
because they don't understand mathematics very
well, generally at a level far below that of classroom
teachers in the schools they serve.
affairs is that the weakest math teachers are usually
the first to embrace the latest education fads, and are
consequently rewarded by principals and other
administrators for their willingness to be innovative.
This kind of “innovation” has a higher priority than
proven effectiveness. The weakest teachers rise
through the administrative ranks in this way. The
least competent teachers end up advising senior
administrators and gain authority over mathematics
programs at all levels.
teacher training.
Would anyone want to leave in charge of writing new state
standards the same people who wrote Washington's current
standards? Does anyone want to leave in charge of textbook
selection the same people who chose TERC, CMP, IMP, and
Core-Plus?
By way of analogy, should the surgeon who consistently
amputates the wrong leg be put in charge of the hospital?
Mathematically competent people must be given actual
decision making power within state and district
bureaucracies.
Recommendations
• Completely rewrite state standards or adopt
already existing high quality standards: California,
Indiana, Massachusetts
• Appoint university mathematicians (not math
education professors) and experienced classroom
teachers (not math administrators) to high level
positions with decision making power over
standards, textbooks, pacing plans, state
assessments, inservices
• Shift control of college teacher education courses
away from colleges of education into subject
matter departments
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