Transcript Core-Plus Mathematics

Core-Plus Mathematics
Curriculum, Instruction,
Assessment
The focus of school mathematics is shifting from a
dualistic mission—minimal mathematics for the
majority, advanced mathematics for a few—to a
singular focus on a common core . . . For all
students.
Everybody Counts
National Research Council
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NCTM Principles and Standards
for School Mathematics
“All students must have access to the highest quality
mathematics instructional programs. A society in which
only a few have the mathematical knowledge needed to
fill crucial economic, political, and scientific roles is not
consistent with the values of a just democratic system
or its economic needs.” (p. 5)
“Expectations must be raised.” (p. 13)
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NCTM Principles and Standards
for School Mathematics
“All students are expected to study mathematics each of
the four years that they are enrolled in high school,
whether they plan to pursue the further study of
mathematics, to enter the workforce, or to pursue other
postsecondary education.” (p. 288)
“Whatever the approach taken, all students learn the
same core material while some, if they wish, can
study additional mathematics consistent with their
interests and career directions.” (p. 289)
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Principles for School Mathematics
The six principles for school mathematics address
overarching themes:
• Equity. Excellence in mathematics education requires
equity—high expectations and strong support for all
students.
• Curriculum. A curriculum is more than a collection of
activities: it must be coherent, focused on important
mathematics, and well articulated across the grades.
• Teaching. Effective mathematics teaching requires
understanding what students know and need to learn
and then challenging and supporting them to learn it
well.
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• Learning. Students must learn mathematics with
understanding, actively building new knowledge from
experience and prior knowledge.
• Assessment. Assessment should support the learning
of important mathematics and furnish useful
information to both teachers and students.
• Technology. Technology is essential in teaching and
learning mathematics; it influences the mathematics
that is taught and enhances students’ learning.
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Core-Plus Mathematics Project
4th-year
Course
Options
3rd-year
Course
2nd-year
Course
1st-year
Course
A Three-Year Core Program
Plus
A Flexible Fourth-Year
Transition to College Mathematics Course
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Design Principles
Three years of mathematical study revolving around a core
curriculum should be required of all secondary school
students. This curriculum should be differentiated by depth
and breadth of treatment of common topics and by the
nature of applications. All students should study a fourth-year
of appropriate mathematics (NCTM, 1989).
Each part of the curriculum should be justified on its own
merits (MSEB, 1990).
Mathematics is a vibrant and broadly useful subject to be
explored and understood as an active science of patterns
(Steen, 1990).
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Design Principles
Problems provide a rich context for developing student
understanding of mathematics (Schoenfeld, 1988;
Schoenfeld, 1992; Heibert, Carpenter, Fennema, Fuson,
Human, Murray, Olivier & Wearne, 1996).
Deep understanding of mathematical ideas includes
connections among related concepts and procedures, both
within mathematics and to the real world (Skemp, 1987).
Computers and calculators have changed not only what
mathematics is important, but also how mathematics should
be taught (Zorn, 1987; Hembree & Dessart, 1992; Dunham &
Dick, 1994).
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Design Principles
Classroom cultures of sense-making shape students
understanding of the nature of mathematics as well as the
ways in which they can use the mathematics they have
learned (Resnick, 1987; Resnick, 1988; Lave, Smith, &
Butler, 1988).
Social interaction (Cobb, 1995) and communication (Silver,
1996) play vital roles in the construction of mathematical
ideas.
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Some Features of the
Core-Plus Mathematics Curriculum
• Broader scope of content to include statistics, probability,
and discrete mathematics each year
• Less compartmentalization, greater integration of
mathematical strands
• Mathematics is developed in context
• Emphasis on mathematical modeling
• Full and appropriate use of graphing calculators
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Some Features of the
Core-Plus Mathematics Curriculum
• Emphasis on active learning—collaborative group
investigations, oral and written communication
• Differentiated applications and extensions of core topics
• Designed to make mathematics accessible to a broader
student population
• Student assessment as an integral part of the curriculum
and instruction
• Flexible fourth-year course for college-bound students
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Integration of Strands
Through Common Topics
• Functions
• Symmetry
• Matrices
• Recursion
• Data analysis and curve fitting
Through Global Themes
• Data
• Representation
• Shape
• Change
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Integration of Strands
Through Habits of Mind
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Search for patterns
Formulate or find a mathematical model
Experiment
Collect, analyze, and interpret data
Make and check a conjecture
Describe and use an algorithm
Visualize
Predict
Prove
Seek and use connections
Use a variety of representations
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Habits of Mind1
General Habits
• Searching for Patterns
• Performing Experiments
• Describing Ideas & Processes
• Playing with Ideas
• Inventing Mathematics
• Visualizing Things, Ideas, Relationships, Processes
• Making & Checking Conjectures
• Guessing
• Turning to Resources
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Adapted from A. Cuoco, E. P. Goldenberg, & J. Mark. “Habits of Mind: An Organizing
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Principle for a Mathematics Curriculum.”
Habits of Mind1
Mathematical Habits
• Classifying
• Analyzing
• Abstracting
• Representing
• Using Multiple Representations
• Algorithmic Thinking
• Visual Thinking
• Making Connections
• Proving
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Adapted from A. Cuoco, E. P. Goldenberg, & J. Mark. “Habits of Mind: An Organizing
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Principle for a Mathematics Curriculum.”
The Core-Plus Mathematics Project
CONTENT STRANDS
Algebra and Functions
Develop student ability to recognize, represent, and solve
problems involving relations among quantitative variables.
Focal Points
• patterns of change
• functions as mathematical models
• linear, exponential, power, logarithmic, polynomial, rational, and
periodic functions
• linked representations—verbal, graphic, numeric, and symbolic
• rates of change and accumulation
• multivariable relations and systems of equations
• symbolic reasoning and manipulation
• structure of number systems
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Statistics and Probability
Develop student ability to analyze data intelligently, recognize
and measure variation, and understand the patterns that
underlie probabilistic situations.
Focal Points
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modeling, interpretation, prediction based on real data
data analysis—graphical and numerical methods
simulation
correlation
probability distributions—geometric, binomial, normal
quality control
surveys and samples
best-fitting data models
hypothesis testing
experimental design
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Geometry and Trigonometry
Develop visual thinking and student ability to construct, reason
with, interpret, and apply mathematical models of patterns in
visual and physical contexts.
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visualization
shape, size, location, and motion
representations of visual patterns
coordinate, transformational, vector, and synthetic
representations and their connections
symmetry, change, and invariance
form and function
trigonometric methods and functions
geometric reasoning and proof
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Discrete Mathematics
Develop student ability to model and solve problems involving
enumeration, sequential change, decision-making in finite
settings, and relationships among a finite number of elements.
Focal Points
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discrete mathematical modeling
recursion
vertex-edge graphs
matrices
optimization and algorithmic problem solving
systematic counting
informatics
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Core-Plus Mathematics
Course 1 Units
Unit 1:
Patterns of Change
Unit 2:
Patterns in Data
Unit 3:
Linear Functions
Unit 4:
Vertex-Edge Graphs
Unit 5:
Exponential Functions
Unit 6:
Patterns in Shape
Unit 7:
Quadratic Functions
Unit 8:
Patterns in Chance
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Core-Plus Mathematics
Course 2 Units
Unit 1:
Functions, Equations, and Systems
Unit 2:
Matrix Methods
Unit 3:
Coordinate Methods
Unit 4:
Regression and Correlation
Unit 5:
Nonlinear Functions and Equations
Unit 6:
Network Optimization
Unit 7:
Trigonometric Methods
Unit 8:
Probability Distributions
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Core-Plus Mathematics
Course 3 Units
Unit 1:
Reasoning and Proof
Unit 2:
Inequalities and Linear Programming
Unit 3:
Similarity and Congruence
Unit 4:
Samples and Variation
Unit 5:
Polynomial and Rational Functions
Unit 6:
Circles and Circular Functions
Unit 7:
Recursion and Iteration
Unit 8:
Inverses of Functions and Logarithms
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Core-Plus Mathematics
Course 4
The mathematical content and sequence of units in
Course 4 allows considerable flexibility in tailoring a
course to best prepare students for various
undergraduate programs.
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Core-Plus Mathematics
Course 4 Units
Unit 1:
Families of Functions
Unit 2:
Vectors and Motion
Unit 3:
Algebraic Functions and Equations
Unit 4:
Trigonometric Functions and Equations
Unit 5:
Exponential Functions, Logarithms, and Equations
Unit 6:
Surfaces and Cross Sections
Unit 7:
Rates of Change
Unit 8:
Counting Methods and Induction
Unit 9:
Binomial Distributions and Statistical Inference
Unit 10: Mathematics of Information Processing and the Internet
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Core-Plus Mathematics
Instructional Model
In-Class Activities
Launch
Full class discussion of a problem situation
and related questions to think about.
Explore
Small group cooperative investigations of
focused problem(s)/question(s) related to
the launching situation.
Share/Summarize
Full class discussion of concepts and
methods developed by different groups
leads to class constructed summary of
important ideas.
Apply
A task for students to complete individually
to assess their understanding.
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Core-Plus Mathematics
Instructional Model
Out-of-Class Activities
On Your Own
Applications
Tasks in this section provide students with
opportunities to use the ideas they
developed in the investigations to model
and solve problems in other situations.
Connections
Tasks in this section help students
organize the mathematics they developed
in the investigations and connect it with
other mathematics they have studied.
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Reflections
Tasks in this section help students think
about what the mathematics they
developed means to them and their
classmates and to help them evaluate their
own understanding.
Extensions
Tasks in this section provide opportunities
for students to explore the mathematics
they are learning further or more deeply.
Review
Tasks in this section provide opportunities
for students to review previously learned
mathematics and to refine their skills in
using that mathematics.
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Core-Plus Mathematics
Assessment Dimensions
Process
Content
Dispositions
Problem Solving
Concepts
Beliefs
Reasoning
Applications
Perseverance
Communication
Representational
Strategies
Confidence
Connections
Procedures
Enthusiasm
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Assessment
Curriculum-Embedded Assessment
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Think About This Situation
Investigation
Summarize the Mathematics
Check Your Understanding
Reports and Presentations
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Questioning
Observing
Student Work and Math Toolkits
On Your Own
Supplementary Assessment Materials
• End-of-Lesson Quizzes
• In-Class Unit Exams
• Take-Home Tasks
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Unit Projects
Portfolios
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ACCESS
Curricular Content
Technology
Pedagogical Approach
Conceptions and Beliefs of Teachers
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Curricular Content
• Multiple strands nurture differing strengths and talents
• Content developed in meaningful, interesting, and
diverse contexts
• Skills are embedded in more global modeling tasks
• Technical language and symbols introduced as the
need arises
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Technology
• Promote versatile ways of dealing with realistic
situations
• Reduce manipulative skill filter
• Offer visual and numerical routes to mathematics that
complement symbolic forms
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Pedagogical Approach
• Lessons as a whole promote discourse as a central
medium for teaching and learning
• Lesson launches and investigations value and build on
informal knowledge
• Investigations promote collaborative learning
• Investigations encourage multiple approaches to tasks
• Summarize the Mathematics questions promote socially
constructed knowledge. Diversity is recognized as an
asset.
• On Your Own tasks accommodate differences in student
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performance, interest, and mathematical knowledge.
The Most Important Aspects
of Teaching Core-Plus Mathematics
as Reported by Teachers
• Creating an atmosphere for risk-taking
• Listening to students
• Planning
• Being able to back off and let the students take responsibility for
their learning
• Checking that students are making valid generalizations
• Seeing the big picture
• Closure
• Teachers must understand the content and the extent that
mathematics is taught developmentally through cooperative
groups and connectively through the strands.
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Classroom Observations
• Atmosphere of cooperation—students are dividing
responsibilities, mediating solutions, and explaining ideas to each
other.
• Teachers are circulating among groups listening to, and guiding,
student thinking.
• Students are active participants, willing to put forth considerable
effort.
• Various approaches to solving problems are encouraged and
accepted.
• Students look for patterns and ways to describe them clearly
rather than just looking for procedures.
• Teachers believe that ALL students can learn mathematics
because they have witnessed and experienced it.
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• Learning is encouraged through students exchanging ideas,
conjecturing, and explaining their reasoning.
• Peer questioning/challenging of thinking and reasoning becomes
common place.
• Student confidence in thinking, reasoning, and problem-solving
ability improves with time and experience.
• Quality of student written work is impressive.
• A variety of assessment tools, including interviews and
observation, is used.
• When using Core-Plus Mathematics with heterogeneously
grouped classes, teachers have indicated that they are unable, in
many instances, to identify students who traditionally would have
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been assigned to a lower level class.
Selected Impact Data
Quantitative Thinking
Core-Plus Mathematics students outperform comparison students
on the mathematics subtest of the nationally standardized Iowa
Tests of Educational Development ITED-Q.
Conceptual Understanding
Core-Plus Mathematics students demonstrate better conceptual
understanding than students in more traditional curricula.
Problem Solving Ability
Core-Plus Mathematics students demonstrate better problem
solving ability than comparison students.
Applications and Mathematical Modeling
Core-Plus Mathematics students are better able to apply
mathematics than students in more traditional curricula.
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Selected Impact Data
Algebraic Reasoning
Core-Plus Mathematics students perform better on tasks of
algebraic reasoning than comparison students. On some
evaluation tests, Core-Plus Mathematics student do as well or
better; on others they do less well than comparison students.
Important Mathematics in Addition to Algebra
Core-Plus Mathematics students perform well on mathematical
tasks involving geometry, probability, statistics, and discrete
mathematics.
National Assessment of Educational Progress (NAEP)
Core-Plus Mathematics students scored well above national
norms on a test comprised of released items from the National
Assessment of Educational Progress.
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District Evaluation Plans
• Comparison studies using eighth grade math achievement as a
baseline
State assessments
Standardized tests whose content both groups had
opportunity to learn
• Student attitude surveys
• Enrollment trends in elective math courses
• Performance in science courses and on science portions of
standardized tests
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Selected Impact Data
Student Perceptions and Attitudes
Core-Plus Mathematics students have better attitudes and
perceptions about mathematics than students in more traditional
curricula.
Performance on State Assessments
The pass rate on the 2004-05 Tenth-Grade Washington
Assessment of Student Learning Mathematics test for 22 sate of
Washington high schools that were in at least their second year
using the Core-Plus Mathematics curriculum was significantly
higher than that of a sample of 22 schools carefully matched on
prior mathematics achievement, percent of students from lowincome families, percent of underrepresented minorities, and
student enrollment.
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College Entrance Exams—SAT and ACT
Core-Plus Mathematics students do as well as, or better than,
comparable students in more traditional curricula on the SAT and
ACT college entrance exams.
College Mathematics Placement Exam
On a mathematics department placement test used at a major
Midwestern university, Core-Plus Mathematics students
performed as well as students in traditional precalculus courses
on basic algebra and advanced algebra subtests, and they
performed better on the calculus readiness subtest.
Performance in College Mathematics Courses
Core-Plus Mathematics students completing the four-year
curriculum perform as well as, or better than, comparable
students in a more traditional curriculum in college mathematics
courses at the calculus level and above.
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