Transcript Document

Advanced Quantitative Reasoning
Mathematics and Statistics for Informed
Citizenship and Decision Making
Gregory D. Foley, PhD
Robert L. Morton Professor of Mathematics Education
Ohio University
Athens, Ohio
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Advanced Quantitative Reasoning (AQR) is a quantitative
literacy course for high school seniors or juniors. Many high
school graduates are not ready for the mathematical demands
of college and work, and never intend to pursue calculus.
The AQR course will provide a model for a post-Algebra II
alternative to Precalculus.
The AQR project is an ongoing effort to—
(a) write, pilot, and hone student text materials;
(b) offer summer institutes to build teacher capacity; and
(c) investigate the nature and level of the student and teacher
learning that takes place.
The AQR course content will incorporate various state and
national recommendations.
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This talk makes a case for an inquiry-based post-Algebra II
capstone mathematics course as the preferred senior year
mathematics option for the majority of high school students.
The proposed course is substantially different from the various
traditional and innovative precalculus courses currently taught
in the United States and has a different set of aims.
The content is drawn from measurement, percent, probability,
statistics, discrete and continuous modeling, geometry in three
dimensions, vectors, and fractals—with strong emphases on
problem solving, reasoning, and communication.
The mathematics is done and learned by students in context
through investigations and projects, and students regularly
report their results.
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The aims of this capstone course are—
• to reinforce, build on, and solidify the student’s working
knowledge of Algebra I, Geometry, and Algebra II
• to develop the student’s quantitative literacy for effective
citizenship, for everyday decision making, for workplace
competitiveness, and for postsecondary education
• to develop the student’s ability to investigate and solve
substantial problems and to communicate with precision
• to prepare the student for postsecondary course work in
statistics, computer science, mathematics, technical fields,
and the natural and social sciences—and
• for students who completed Algebra I in the 8th grade, to
prepare them to study AP Statistics, AP Computer Sciences,
or Precalculus in their senior year of high school
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Several interacting forces create the need for a
post-Algebra II alternative to Precalculus.
• “Only about 25% of high school graduates take precalculus
in high school, even though over 60% enroll in some form
of postsecondary education” (Steen, 2006, p. 10).
• “Only a small percentage of students who take precalculus
ever go on to take calculus, and many who do are not well
prepared and never complete the next course” (Baxter
Hastings et al., 2006, p. 1).
• “Perhaps the worst thing that can happen to a student at the
end of his or her secondary mathematics preparation is to
enter college not having studied mathematics after a lapse
of a year or more” (Seeley, 2004, p. 24).
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Related initiatives and reports
• Standards for School Mathematics (NCTM, 1989, 2000)
• NSF-supported curriculum development projects
• American Diploma Project Creating a High School Diploma
That Counts (Achieve, Inc., 2004)
• A Fresh Start for Collegiate Mathematics: Rethinking the
Course Below Calculus (MAA, Baxter Hastings et al., 2006)
• Standards for College Success: Mathematics and Statistics
(College Board, 2006)
• Current Practices in Quantitative Literacy (Gillman, 2006)
• Math Takes Time position statement (NCTM, 2006)
• Guidelines for Assessment and Instruction in Statistics
Education (GAISE) report (Amer. Stat. Ass’n, 2007)
• Modeling & Quantitative Reasoning (Ohio Dep’t of Ed, 2007)
• Advanced Mathematical Decision Making (UT Dana, 2008)
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NCTM Math Takes Time (2006):
• Every student should study mathematics every year through
high school, progressing to a more advanced level each year.
All students need to be engaged in learning challenging
mathematics.
• At every grade level, students must have time to become
engaged in mathematics that promotes reasoning and fosters
communication.
• Evidence supports the enrollment of high school students in a
mathematics course every year, continuing beyond the
equivalent of a second year of algebra and a year of geometry.
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The proposed course is for the majority of students
who do not intend to pursue college majors or careers
that require knowledge of calculus.
The need for such a course has been recognized—
• in North Carolina since 2001,
• recently in Kentucky, Ohio, Texas,
Washington, and Wyoming,
• and elsewhere across the United States.
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Advanced Quantitative Reasoning content
1. Data Analysis, Probability, & Statistics
2. Discrete Mathematics
3. Advanced Functions & Modeling
4. Advanced Topics in Geometry
5. “Numbers Everywhere”: a focus on uses of numbers as
measurements, metrics, indices, and identification codes.
These will be the topics for teacher professional development.
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Advanced Quantitative Reasoning course outline
Part A. Explorations, Activities, Investigations, with increasingly
involved small projects and presentations (30–32 weeks)
1. Numerical Reasoning—with tone setting (6–8 weeks)
2. Statistical Reasoning (5–7 weeks)
3. Discrete and Continuous Modeling (9–12 weeks)
4. Spatial Reasoning (6–9 weeks)
— “Numbers Everywhere” vignettes throughout —
Part B. Course Research Project (4–6 weeks)
5. Project Planning
6. Project Implementation and Report Writing
7. Public Presentation of Project Results
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Unit 1. Numerical Reasoning (with tone setting)
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Percentages used as fractions, to describe change, and to
show comparisons, while setting course expectations for
collaboration, investigation, and communication
(e.g., sale prices, inflation, cost of living index and other
indices, tax rates, and medical studies)
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Compound percents used in financial applications
(e.g., savings and investments, loans, credit cards,
mortgages, and federal debt)
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Combinatorics and Probability
(e.g., insurance, lottery, random number generation,
weather forecasting, and probability simulations)
— “Numbers Everywhere” thread established —
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Unit 2. Statistical Reasoning: analyzing variability
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Understanding the statistical process: formulating a
question, collecting and analyzing data, and interpreting
results
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Using appropriate summary statistics and formulating
reasonable conclusions
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Identifying bias and abuses of statistics
(e.g., margin for error, sampling bias within surveys
and opinion polls, correlation versus causation)
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Unit 3. Discrete and Continuous Modeling
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Social choice and decision making
Recurrence relations, including linear difference equations
Direct proportion and linear models
Step and piecewise models
Exponential and power functions
Logarithmic scaling models and logarithmic re-expression
Periodic functions include sinusoidal trigonometric functions
Logistic functions
(e.g., unit conversions, straight line depreciation, simple
interest, population growth, radioactive decay, pH, Richter
scale, inflation, depreciation¸ periodic doses, sound waves,
sunlight per day, bouncing balls, oscillating springs, spread of
a rumor, spread of a disease, chemical reactions)
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Unit 4. Spatial Reasoning
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Vertex-edge graphs
Connectivity matrices
Visual models for functions of two variables
Vectors as representational tools
Polar coordinates
Fractal geometry
(e.g., decision trees, spanning trees, routing and production
problems, weather maps, topographic maps, forces,
velocities, displacements, translations, latitude, longitude,
polar maps, measuring an island coast line, the length of a
meandering stream, area of a square leaf with holes in a
fractal pattern)
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Illustrative Examples
1. Numerical Reasoning: Developing amortization
schedules using a spreadsheet
2. Statistical Reasoning: Developing and carrying
out a small statistical study
3. Discrete and Continuous Modeling: Exploring
patterns and developing models for the
populations over time for Florida and
Pennsylvania
4. Spatial Reasoning: Interpreting USA Today
weather maps
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A series of related funded projects
Projects already funded
1. Ohio Board of Regents Improving Teacher Quality grant for
professional development in Probability & Statistics (2008–09)
2. SEOCEMS grant for initial student and teacher materials
development and for research preparation (Summer 2008)
3. Ohio Department of Education grant for teacher professional
development throughout Ohio (2008–2010)
Proposal under development
4. NSF DR-K12 curriculum research and development in
Ohio, Kentucky, and Texas (2009–10 through 2013–14)
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Issues to be addressed
• Staffing
• Teacher preparation in statistics, discrete mathematics,
modeling, and advanced topics in geometry
• Teacher preparation in inquiry-based mathematics, creative
uses of technology, and project-based instruction
• Teacher professional development in these same areas
• Text materials with the appropriate content at the
appropriate level with investigations and projects
• Curriculum development, pilot testing, and implementation
• Roles of technology and needed technology resources
• Supplementary materials: on-line and in periodicals
• Length of final research project
• Differentiated instruction
• Others???
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Selected curricular resources
Andersen, J., & Swanson, T. (2005). Understanding our
quantitative world. Mathematical Association of America.
Blocksma, M. (2002). Necessary numbers. Portable Press.
COMAP. (2003). For all practical purposes. W. H. Freeman.
Crisler, N., Fisher, P., & Froelich, G. (2000). Discrete
mathematics through applications (2/e). W. H. Freeman.
Demana, F., Waits, B. K., Foley, G. D., & Kennedy, D. (2007).
Precalculus: Graphical, numerical, algebraic (7/e). Pearson.
Sevilla, A., & Somers, K. (2007). Quantitative reasoning: Tools
for today’s informed citizen. Key College Publishing.
Souhrada, T. A., & Fong, P. W. (Eds.). (2006a, b). SIMMS
integrated mathematics, Levels 3 & 4 (3/e). Kendall/Hunt.
Yoshiwara, K., & Yoshiwara, B. (2007). Modeling, functions,
and graphs (4/e). Thomson Brooks/Cole.
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Advanced Quantitative Reasoning
Mathematics and Statistics for Informed
Citizenship and Decision Making
Gregory D. Foley
Ohio University
Athens, Ohio
Email: [email protected]
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