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What do you want to learn most during this workshop?
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Favorite Summer Activities
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Mathematical Expectations of Colleges
Three-quarters of college majors require mathematics:
about a quarter in business
about a quarter in engineering or the physical sciences
about a quarter in various social sciences or education
(Usiskin, 1988)
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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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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 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). 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).
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Design Principles Social interaction (Cobb, 1995) and communication (Silver, 1996) play vital roles in the construction of mathematical ideas. 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).
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Design Principles
Small-group cooperative learning environments:
encourage more female participation in the mathematics classroom (Wisconsin Center for Education Research, 1994)
encourage a variety of social skills that appear particularly conducive to the learning styles of females and underrepresented minorities (Oakes, 1990; Leder, 1992
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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
1 Adapted from A. Cuoco, E. P. Goldenberg, & J. Mark. “Habits of Mind: An Organizing Principle for a Mathematics Curriculum.”
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Habits of Mind 1
Mathematical Habits
Classifying
Analyzing
Abstracting
Representing
Using Multiple Representations
Algorithmic Thinking
Visual Thinking
Making Connections
Proving
1 Adapted from A. Cuoco, E. P. Goldenberg, & J. Mark. “Habits of Mind: An Organizing Principle for a Mathematics Curriculum.”
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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
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. Focal Points
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
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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Curriculum-Embedded Assessment
Think About This Situation • Questioning
Investigation • Observing
Summarize the Mathematics • Student Work and Math Toolkits
Check Your Understanding • On Your Own
Reports and Presentations
Supplementary Assessment Materials
End-of-Lesson Quizzes • Unit Projects
In-Class Unit Exams • Portfolios
Take-Home Tasks
Assessment
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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 performance, interest, and mathematical knowledge.
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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 been assigned to a lower level class.
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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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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. Selected Impact Data
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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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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 low-income families, percent of underrepresented minorities, and student enrollment. Selected Impact Data
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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.