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Advanced Quantitative ReasoningMathematics and Statistics for Informed

Citizenship and Decision Making

Gregory D. Foley, PhDRobert 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) • 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 funded1. 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 development4. 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 ReasoningMathematics and Statistics for Informed

Citizenship and Decision Making

Gregory D. FoleyOhio University

Athens, Ohio

Email: foleyg@ohio.edu