Washington

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Washingtons Math Standards

• David Klein
• Professor of Mathematics
• California State University, Northridge

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Why do standards matter?

• goal posts for teaching and learning
• coherence across grade levels
• determine the content and emphasis of tests
• influence the selection of textbooks
• form the core of teacher education programs

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The State of State Math Standards 2005
Fordham Foundation

• Co-authors of the Fordham Foundation Report
• 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

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Whats Wrong with Washington's Standards?
Fordham Foundation grade F
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
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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).

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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.
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First Grade Sample ProblemA 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.Grade 9/10 sample problem asks if it
is reasonable to 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.
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CalculatorsTechnology should be available and
used throughout the K12mathematics curriculum.
In the early years, students can usebasic
calculators to examine and create patterns of
numbers. Calculators introduced in 1st
grade2nd grade standard Solve problems
involving addition and subtraction with two or
three digit numbers using a calculator and
explaining procedures used.
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FractionsIntroduced for the first time in 4th
gradeExplain 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.
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Fractions Grade 6Explain the meaning of
multiplying and dividing non-negative 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
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PatternsWhat 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
Instructionwww.k12.wa.us/curriculumInstruct/mathe
matics/default.aspx
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Patterns 6th Grade

• 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
  operations including addition, subtraction,
  multiplication, or division.
• Use rules for generating number patterns (e.g.,
  Fibonacci sequence, bouncing ball) to model
  real-life situations.
• Use technology to generate patterns based on two
  arithmetic operations. Supply missing elements in
  a pattern based on two operations.

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More Patterns, 6th Grade

• 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

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6th Grade WASL

• Karen made a triangle out of number tiles.
  She used a rule to create the pattern in the
  number tiles.
• Extend the pattern to complete the next row of
  the triangle.
• Describe the rule you used to extend the
  pattern.

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Why are most state standards, including
Washington's, so bad?

• 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).

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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.
• All students should have access to
  calculators... throughout their school
  mathematics program
• decreased emphasis on...performing paper and
  pencil calculations with numbers of more than two
  digits.
• de-emphasis of calculus

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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
  exact forms of answers

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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.
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NCTM-Aligned Books
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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.

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NCTM Reply on Textbooks

• 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.

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TERC 5th grade fractions
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
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Math Wars Points of Emphasis

• Mathematical Content
• Skills-based
• cohesion, clarity
• proof
• Classical tradition
• whole class instruction
• teacher centered
• model university
• parents, mathematicians, academics

• Pedagogy
• learn skills as needed for real world problems
• discovery learning
• learning styles
• Romantic tradition
• small group learning
• child centered
• model kindergarten
• colleges of education, school administrators,
  corporate leaders

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Mathematical CompetenceMathematical 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.
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The mechanism leading to this paradoxical state
of 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.
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The result is bad standards, bad books, bad
tests, and bad 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.
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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