Advanced Quantitative Reasoning Mathematics and Statistics for Informed Citizenship and Decision Making

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Advanced Quantitative ReasoningMathematics and
Statistics for InformedCitizenship 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 fractalswith 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
  students working knowledge of Algebra I,
  Geometry, and Algebra II
• to develop the students quantitative literacy
  for effective citizenship, for everyday decision
  making, for workplace competitiveness, and for
  postsecondary education
• to develop the students 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 sciencesand
• 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.
  Assn, 2007)
• Modeling Quantitative Reasoning (Ohio Dept 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 (3032 weeks)
• Numerical Reasoningwith tone setting (68 weeks)
• Statistical Reasoning (57 weeks)
• Discrete and Continuous Modeling (912 weeks)
• Spatial Reasoning (69 weeks)

Numbers Everywhere vignettes throughout

• Part B. Course Research Project (46 weeks)
• Project Planning
• Project Implementation and Report Writing
• Public Presentation of Project Results

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Unit 1. Numerical Reasoning (with tone setting)

• 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)
• Compound percents used in financial applications
  (e.g., savings and investments, loans,
  credit cards, mortgages, and federal debt)
• 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

• Understanding the statistical process
  formulating a question, collecting and analyzing
  data, and interpreting results
• Using appropriate summary statistics and
  formulating reasonable conclusions
• 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

• 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

• 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
• Ohio Board of Regents Improving Teacher Quality
  grant for professional development in Probability
  Statistics (200809)
• SEOCEMS grant for initial student and teacher
  materials development and for research
  preparation (Summer 2008)
• Ohio Department of Education grant for teacher
  professional development throughout Ohio
  (20082010)

• Proposal under development
• NSF DR-K12 curriculum research and development in
  Ohio, Kentucky, and Texas (200910 through
  201314)

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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 todays 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 ReasoningMathematics and
Statistics for InformedCitizenship and Decision
Making

• Gregory D. Foley
• Ohio University
• Athens, Ohio
• Email foleyg_at_ohio.edu