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### 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

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)

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)

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.

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.

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).

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).

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.

Some Features of theCore-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

Some Features of theCore-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

- Through Common Topics
- Functions
- Symmetry
- Matrices
- Recursion
- Data analysis and curve fitting

- Through Global Themes
- Data
- Representation
- Shape
- Change

- Through Habits of Mind
- 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

- 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

- 1 Adapted from A. Cuoco, E. P. Goldenberg, & J. Mark. “Habits of Mind: An Organizing Principle for a Mathematics Curriculum.”

- 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.”

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

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

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

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

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

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

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

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.

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

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.

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.

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.

Assessment Dimensions

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

- 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

- Promote versatile ways of dealing with realistic situations
- Reduce manipulative skill filter
- Offer visual and numerical routes to mathematics that complement symbolic forms

- 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.

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.

- 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.

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.

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.

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.

- 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

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.

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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