Transcript Document

1
2
Why?
3
Why Identify Focal Points?
• Address long lists of state learning expectations
• Address “mile wide, inch deep” math curriculum
• Identify the mathematics that should be the focus
of instruction and student learning, preK-8
• Begin the discussion of appropriate curricular
expectations
• Identify key mathematical ideas all others build on
4
5
Number of 4th-Grade Learning Expectations
per State by Content Strand
Number
&
Operation
Geometry
Measurement
Algebra
Data
Analysis,
Probability &
Statistics
Total Number of
Learning
Expectations
California
16
11
4
7
5
43
Texas
15
7
3
4
3
32
New York
27
8
10
5
6
56
Florida
31
11
17
10
20
89
Ohio
15
8
6
6
13
48
Michigan
37
5
11
0
3
56
New Jersey
21
10
8
6
11
56
North Carolina
14
3
2
3
4
26
Georgia
23
10
5
3
4
45
Virginia
17
8
11
2
3
41
Reys, et al., 2006
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What?
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Principles
Content Standards
•
•
•
•
•
•
•
•
•
•
•
Equity
Curriculum
Teaching
Learning
Assessment
Technology
Number/Operations
Algebra
Geometry
Measurement
Data/Probability
Process Standards
•
•
•
•
•
Problem Solving
Reasoning
Communication
Connections
Representation
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NCTM Curriculum Principle
• A curriculum is more than a collection of
activities: it must be
– coherent
– focused on important mathematics
– well articulated across the grades
Principles and Standards for School Mathematics, page 14
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NCTM Curriculum Principle
“…a well-articulated curriculum gives teachers
guidance regarding important ideas or major
themes, which receive special attention at different
points in time. It also gives guidance about the
depth of study warranted at particular times and
when closure is expected for particular skills or
concepts.”
Principles and Standards, p. 16
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What Are Curriculum Focal Points?
• Important mathematical topics for each grade
level, preK-8
• Cohesive clusters of related ideas, concepts,
skills, and procedures that form the foundation
for higher-level mathematics
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What Are Curriculum Focal Points?
• More than a single objective, standard,
expectation, or indicator
• Not discrete topics for teachers to present and
check off as mastered by students
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The Product:
Process Standards
Introductory statement for each level,
PreK-8:
“It is essential that these focal points be
addressed in contexts that promote problem
solving, reasoning, communication, making
connections, and designing and analyzing
representations.”
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14
15
10 x 8
4x8
14 x 8 = (10 x 8) + (4 x 8)
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The Product:
Curriculum Focal Points
• Three per grade level, preK-8
• Often represent multiple content strands
• Describe the majority of instruction for a
specific grade level
• Taken together across grade levels,
provide the major components of a
mathematically sound, coherent and
cohesive preK-8 curriculum
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The Product: Connections to the
Curriculum Focal Points
• Provide meaningful contexts for the focal
points
• Identify connections between strands and
across grade levels
• Round out a well-balanced curriculum
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The Process:
Incorporating a Research Base
• Content- and pedagogy-related studies
(found in publications such as JRME, AERJ,
and those from NAEYC)
• National and international measures of
students’ mathematical proficiencies
(e.g., NAEP, TIMSS, PISA)
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How?
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Curriculum Focal Points and State
and District Leaders
• As a framework for future development of
mathematics curriculum
• To identify grade-level targets
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Curriculum Focal Points and Teachers
• To design instruction around the question,
“What are the most important ideas at my
grade level?”
• To provide information about how ideas at
one grade level fit with the important ideas in
previous and following grades
• To prioritize uses of activities, assessments
and other published materials
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Curriculum Focal Points and
Publishers
As an example for guiding the next
generation of instructional materials and
related assessments
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Curriculum Focal Points and
Teacher Educators
To organize preservice and inservice
education for developing teachers’
knowledge of critical mathematics
understandings across the grades
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Who did this?
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Participation
• Writing group
– Mathematicians
– Mathematics educators
– Teachers
• Outside reviewers
– Mathematicians and mathematics educators
– Teachers and supervisors
– Policymakers
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Curriculum Focal Points:
What’s New
• Priorities - focus
• Grade-by-grade descriptions
• Descriptive clusters of content
• More clarification
• Connections
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Curriculum Focal Points
What’s Not New
• Alignment with Principles and Standards for
School Mathematics, particularly the
Curriculum Principle
• Well-balanced curriculum
• Strong attention to number and operations
• Commitment to problem solving, processes and
content
• Understanding math, doing math, using math
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Then What?
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September 12, 2006
Arithmetic Problem
New Report Urges Return to Basics in Teaching Math
By JOHN HECHINGER
Critics of ‘Fuzzy’ Methods Cheer Educators’ Findings; Drills
Without Calculators
The nation’s math teachers, on the front lines of a 17-year
Curriculum war, are getting some new marching orders: Make sure
students learn the basics.
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September 21, 2006
Latest ‘new math’ idea gets back to the basics
By Stephanie Banchero
1202!!!
For nearly two decades, a battle has raged over the best
ways to teach elementary and high school math.
On one side sit fundamentalists, who prefer oldfashioned drilling and a focus on the basics. On the
other side are the so-called ‘new math’ proponents, who
care more about understanding the concepts than
performing the calculations.
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April 13, 2000
Math Teachers Back Return Of Education in Basic
Skills
By ANEMONA HARTOCOLLIS
In an important about-face, the nation's most influential
group of mathematics teachers announced yesterday
that it was recommending, in essence, that the arithmetic
be put back into mathematics, urging teachers to
emphasize the fundamentals of computation rather than
focus on concepts and reasoning.
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• Children should master the basic facts of arithmetic that
are essential components of fluency with paper-pencil
and mental computation and with estimation.
• It is important for children to learn the sequence of steps
– and the reason for them – in the paper-and-pencil
algorithms used widely in our culture.
PreK-4 – Curriculum and Evaluation Standards, NCTM, 1989, p.47
34
• Knowing basic number combinations – the single digit addition and
multiplication pairs and their counterparts for subtraction and
division – is essential.
• Equally essential is computational fluency – having and using
efficient and accurate methods for computing. Fluency may be
manifested in using a combination of mental strategies and jottings
on paper or using an algorithm with paper and pencil, particularly
when the numbers are large, to produce accurate results quickly.
Regardless of the particular algorithm used, students should be able
to explain their method, understand that many methods exist, and
see the usefulness of methods that are efficient, accurate, and
general.
Number & Operations, Principles and Standards for School Mathematics,
NCTM, 2000, p. 32
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And then….
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• States who have met* or will meet to consider using the
Curriculum Focal Points to assist in revising their state
standards:
–
–
–
–
–
–
–
–
–
–
–
–
Florida
Maine
North Carolina
South Carolina
Minnesota
New York
Pennsylvania
Mississippi
Tennessee
Utah
Maryland
District of Columbia
As of December 6, 2006
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Other Initiatives
•
•
•
•
•
•
•
•
•
National Math Panel Presentation
Capitol – Senate and House Aides
AMS, MAA – October and January
Major Publishers
CBMS Presentation - December
Department of Education – MSP Meetings
Brookings Institution Meeting*
Curriculum Center Meeting – February
Others…
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Concerns
• Confusion – Concept vs Content
• Will states and school districts “drop”
topics?
• When will the tests change?
• Push back internally
– Why this topic at this grade level (see above)?
– Should NCTM have taken more time?
–…
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Comic relief?
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The Good Stuff…
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Sunday, Nov. 19, 2006
How to End the Math Wars
We have a new formula for teaching kids. Don't let
ideology ruin it this time
By CLAUDIA WALLIS
American education is every bit as polarized, red and blue, as American politics. On the crimson, conservative end of the spectrum are
those who adhere to the back-to-basics credo: Kids, practice those spelling words and times tables, sit still and listen to the teacher; school
isn't meant to be fun--hard work builds character. On the opposite, indigo extreme are the currently unfashionable "progressives," who
believe that learning should be like breathing--natural and relaxed, that school should take its cues from a child's interests. As in politics,
good sense lies toward the center, but the pendulum keeps sweeping sharply from right to left and back again. And the kids end up
whiplashed.
Since the Reading Wars of the '90s, the U.S. has largely gone red. Remember the Reading Wars? In the '80s, educators embraced "whole
language" as the key to teaching kids to love reading. Instead of using "See Dick and Jane run" primers, grade-school teachers taught
reading with authentic kid lit: storybooks by respected authors, like Eric Carle (Polar Bear, Polar Bear). They encouraged 5- and 6-yearolds to write with "inventive spelling." It was fun. Teachers felt creative. The founders of whole language never intended it to displace the
teaching of phonics or proper spelling, but that's what happened in many places. The result was a generation of kids who couldn't spell,
including a high percentage who had to be turned over to special-ed instructors to learn how to read. That eventually ushered in the current
joyless back-to-phonics movement, with its endless hours of reading-skill drills. Welcome back, Dick and Jane.
Now we're into the Math Wars. With American kids foundering on state math exams and getting clobbered on international tests by their
peers in Singapore and Belgium, parents and policymakers have been searching for a culprit. They've found it in the math equivalent of
whole language--so-called fuzzy math, an object of parental contempt from coast to coast. Fuzzy math, properly called reform math, is the
bastard child of teaching standards introduced by the National Council of Teachers of Mathematics (N.C.T.M) in 1989. Like whole
language, it was a sensible approach that got distorted into a parody of itself. The reform standards, for instance, called for teaching the
uses of a calculator and estimation, but some educators took that as a license to stop drilling the multiplication tables, skip past long
division and give lots of partial credit for wrong answers. "Some of the textbooks and materials were absolutely hideous," says R. James
Milgram, a professor of mathematics at Stanford.
Adding to the math morass was the fact that 49 states (all but Iowa) devised their own math standards, with up to 100 different goals for
each grade level. Textbook publishers responded with textbooks that tried to incorporate every goal of every state. "There are some 700page third-grade math books out there," says N.C.T.M.'s current president Francis (Skip) Fennell, professor of education at Maryland's
McDaniel College.
Now the N.C.T.M. itself has come riding to the rescue. In a notably slim document, it has identified just three essential goals, or "focal
points," for each grade from pre-K to eighth, none of them fuzzy, all of them building blocks for higher math. In fourth grade, for instance,
the group recommends focusing on the quick recall of multiplication facts, a deep understanding of decimals and the ability to measure and
compute the area of rectangles, circles and other shapes. "Our objective," says Fennell, "is to get conversations going at the state level
about what really is important." In recent weeks, that's begun to happen. Florida and Utah and half a dozen other states are talking about
revising their math standards to match the pared-down approach. That pleases academic mathematicians like Milgram, who notes that this
kind of instruction is what works in math-proficient nations like Singapore.
So do we have a solution to the national math problem? We certainly have the correct formula. The question is, Can we apply it? Already
the N.C.T.M.'s focal points are being called a back-to-basics movement, another swing of the ideological pendulum rather than a fresh look
at what it would take to get more kids to calculus by 12th grade. If the script follows that of the Reading Wars, what comes next will be
dreary times-tables recitals in unison, dull new books that fail to inspire understanding, and drill, drill, drill, much like the unhappy scenes in
many of today's "Reading First" classrooms. And that would be just another kind of math fiasco--of the red variety. Kids will learn their
times tables for sure, but they'll also learn to hate math.
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November 27, 2006
Problem solved
School surveys show that more American students are taking math courses such as algebra and calculus –
but what are they learning? A kind of phony debate has sprung up about whether they need more basics,
such as multiplication and long division, or more so-called creative applications such as problem solving. The
sensible answer, according to the nation's math teachers, is both. If more students understand the basics,
they can apply that knowledge to solving complex problems. And they can also help keep America globally
competitive. The importance of understanding basic math shouldn't be in question. But some parents and
school districts think there was too much emphasis on so-called reform math after a 1989 report by the
Influential National Council of Teachers of Mathematics seemed to encourage students to tackle math
problems - and deal with their intimidation by the subject - through pictures, writing or other methods. But
two more recent NCTM reports, in 2000 and earlier this all, try to give more clarity to important principles and
standards related to math instruction, particularly as more and more goals, assessments and other layers of
accountability have been added by federal and state education officials.
NCTM's most recent report in September rightly re-emphasizes "coherence" in math curriculums, outlining
essential concepts and skills that students should be able to master from pre-kindergarten through eighth
grade. The idea is to encourage states to refocus attention on fundamental and common lessons and skills,
from whole numbers to linear equations, that highly mobile students will understand wherever they attend
school. Maryland State Department of Education officials think the state's math curriculum already strikes a
good balance. But they are wisely conducting a periodic re-examination with school district math
coordinators that should be completed by the end of the year. Since American students generally continue
to lag behind their foreign peers on international math tests, rigorous state curriculum reviews are certainly
appropriate. If America is going to produce enough scientists, engineers and math teachers, then improving
math proficiency at all grade levels, focusing on the basics and problem solving, is key.
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Local Schools to Study Whether Math -- Topics = Better Instruction
By Daniel de Vise
Washington Post Staff Writer
Tuesday, December 5, 2006; A01
Advocates of new math and old math, back-to-basics math and "fuzzy" math might be shelving their
differences to collectively focus on what many consider a more pressing problem: too much math.
Maryland math leaders meet today -- and D.C. math educators gather tomorrow – to discuss
Curriculum Focal Points, a new document from the influential National Council of Teachers of
Mathematics that could profoundly influence math instruction in the region and nationwide. It says the
typical state math curriculum runs a mile wide and an inch deep, resulting in students being introduced
to too many concepts but mastering too few, and urges educators to slim down those lessons.
Some are calling Focal Points the most significant publication in the field since the 1980s. R. James
Milgram, a Stanford University math professor who is among the harshest critics of U.S. math
instruction, said the 41-page report aligns teaching "with what is being done with unbelievable
success" in other countries.
What lies ahead, all agree, is a comparison of Focal Points and state math curricula. Critics of math
education hope that process will lead some states to delete entire sections of their lesson plans.
Most of the topics listed in Maryland's math curriculum and Virginia's math standards can be found in
the Focal Points document. But Focal Points is far more selective in identifying the essential math
topics for each grade.
"Focal Points is saying, 'Teach a few things, and teach them well,' " Bliss said.
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Questions…
• Can curriculum/standards designed around a few
key ideas structure a comprehensive program?
• Can assessments focus on priorities and
problem solving?
• How might textbooks/materials look different if
structured around focal points?
• How can state/federal policies best support rich,
deep appropriate mathematics for every student?
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The Goal: Curriculum Focal Points and
Improved Mathematics Education
• Guidance for schools and states in the design of
curricula and assessment that target the most
important topics
• Focus for teachers that gives sufficient time for
students to understand concepts and develop and
apply skills necessary for future mathematics
• Clear direction for students and parents on the
importance of deep understanding of particular
topics at each grade level
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Your Questions?
49
50
• Children should master the basic facts of arithmetic that
are essential components of fluency with paper-pencil
and mental computation and with estimation.
• It is important for children to learn the sequence of steps
– and the reason for them – in the paper-and-pencil
algorithms used widely in our culture.
PreK-4 – Curriculum and Evaluation Standards, NCTM, 1989, p.47
51
• Knowing basic number combinations – the single digit addition and
multiplication pairs and their counterparts for subtraction and
division – is essential.
• Equally essential is computational fluency – having and using
efficient and accurate methods for computing. Fluency may be
manifested in using a combination of mental strategies and jottings
on paper or using an algorithm with paper and pencil, particularly
when the numbers are large, to produce accurate results quickly.
Regardless of the particular algorithm used, students should be able
to explain their method, understand that many methods exist, and
see the usefulness of methods that are efficient, accurate, and
general.
Number & Operations, Principles and Standards for School Mathematics,
NCTM, 2000, p. 32
52
•
•
Number and Operations and Algebra: Developing Quick Recall of
Multiplication Facts and Related Division Facts and Fluency with
Whole Number Multiplication
Students use understandings of multiplication to develop quick recall of the
basic multiplication facts and related division facts. They apply their
understanding of models for multiplication (i.e., equal-sized groups, arrays,
area models, equal intervals on the number line), place value, and
properties of operations, in particular the distributive property, as they
develop, discuss, and use efficient, accurate, and generalizable methods to
multiply multi-digit whole numbers. They select and accurately apply
appropriate methods to estimate products and mentally calculate products
depending upon the context and the numbers involved. They develop
fluency with efficient procedures, including the standard algorithm, for
multiplying whole numbers; understand why the procedures work based on
place value and properties of operations; and use them to solve problems.
Curriculum Focal Points for PreK through Grade 8 Mathematics, NCTM,
2006, p.16
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