Students need a foundation in mathematics to support their academic and workplace success. Quantitative literacy is required because the skills of interpreting data and reasoning in general are needed to deeply understand the arguments and information presented in today's world. Workplace mathematics will help students in broad groups of employment situations as well as in classes that apply to specific fields. Finally students who develop a mathematical habit of mind will regularly apply mathematics to find patterns and develop conjectures. (Habit of mind? confusing...habit of thought, perhaps) To study data and develop conjectures, students will need to overcome their anxiety about mathematics and challenge themselves to use the mathematical tools they have learned. Quantitative Literacy
Quantitative literacy includes mental and technology aided computational skills and knowing when computation is appropriate, knowing when results are reasonable, as well as thinking about the world in terms of numbers and models. On a more advanced level quantitative literacy can include making and supporting arguments using numbers and mathematics.

Some core topics in quantitative literacy are listed in bold below. Sample topics within that area are given to help clarify what the core topic includes.

NUMBER SENSE
Students should recognize the value of small numbers, including significance and precision.

Students should be able to estimate products, sums, differences, and quotients of two numbers to determine the magnitude of the answer and to check the reasonableness of a computation.

Students should recognize how much precision a number has or needs to have. Students should recognize when rounded numbers are being used in contexts, such as reports in the media. (There are 110,000 people below the poverty line in the city, for instance.)

Use of Technology: Students should understand how calculators and spreadsheets use order of operations, should understand the significance of calculator results, and how to communicate these results within the precision of the problem.

FINANCE
Students should understand compound interest and simple interest, and the difference between them, as well as the effect that interest has on credit card debt.

Students should understand the nonlinear nature of compounded interest rates such as the future value of money.

Students should understand formulae that can be used to compute payments and know how to use technology such as spreadsheets to create amortization schedules.

Students should recognize the absolute size of large numbers like millions, billions and trillions and their relative size to one another.

STATISTICS
Students should understand how probability and expected value can be used to evaluate policies and to make decisions about what policies to make.

Students should be able read and make decisions based upon data fromline graphs, bar graphs, histograms, box and whisker plots, scatterplots and pie charts.

Students should recognize when graph scales can lead to misinterpretation of data, and how to correctly construct graph scales for basic situations.

Students should recognize the differences between measures of central tendency, which measure is most appropriate to use and how to use multiple measures for better communication.

Students should understand percentile rank.

Students should know reasons why statistical arguments can be faulty such as those taken from [2]
including especially these common sources of error: sampling error; misinterpreting averages or probabilities

RATIONAL NUMBERS
Students should recognize fraction, decimal, and percent equivalencies for common fractions like halves, fourths, fifths and tenths.

Students should be able to make mental calculations involving common percentages and fractions; like 10% of a number, 1/2, 1/3 and 1/4 of a number, 110% of a number, 15% - 20% tips.

Students should be able to calculate a percent increase, or percent decrease.

Students should know the relation between base and percent. For instance the state of Illinois has more Spanish-speakers at home than New Mexico despite over 25% of New Mexicans having Spanish as a first language because of Illinois's overall population.

Students should be able to graph y = 1/x and verify points on the curve.

RATE AND PROPORTION

Students should be able to compute and apply the average rate of change from tabular data and include the appropriate units.

Students should be able to make calculations such as those found in utility bills.

Students should be able to estimate a total given a unit rate and the number of units.

Students should know when it is more appropriate to use the actual number and when it is more appropriate to use a rate. For instance when comparing crime statistics between a medium sized and a large city the rate per 10,000 is probably more useful than the number of crimes.

MEASUREMENT

Students should know when to use length, area and volume, and understand the units involved with each.

Students should be able to explain the units used in rates and make conversions between units.

Students should be able to estimate lengths and volumes in US Standard and metric units and use these in calculation.

Students should recognize basic two- and three-dimensional shapes involving rectangular, triangular, spherical and conical components.

ALGEBRA

Students should be able to use variables as placeholders as in formulas.

Students should know what a solution to an equation is.

Students should understand scientific notation.

Students should recognize the difference between discrete and continuous functions, and understand when each type is more appropriate.

Students should be able to judge when to use an algorithm and be able to apply the steps sequentially.

? Students should be competent at problem solving ... writing algebraic statements for descriptions of problems

? Students should be able to evaluate expressions using the order of operations, including exponents and parentheses

? Students should understand the difference between a variable and a constant

Mathematics in the Workplace
In addition to general quantitative literacy, many fields of study require an understanding of statistics and modeling; and require enough algebra to understand, use and create spreadsheets.

ADDITIONAL STATISTICS SPECIFIC TO THE WORKPLACE

Students should be able to read data presented in a table and write data into a table. Students should recognize that matrices (arrays) can be used to manipulate data as well as store data.

Students should have a basic understanding of reading graphs and tables as detailed in the Quantitative Literacy section above.

MODELING
Students should understand linear, exponential, power and logistic growth as described in the Math as a Tool to Understand the World Section below.

Students should be able to design an experiment that collects appropriate data in order to find a reasonable model and then use that model to make predictions and/or observations.

Students should understand the meaning of variable, including formulas and evaluation. Students should be able to use spreadsheet cell references as variables.

Students should be able to use spreadhshhets to investigate scenarios. Math as a Tool to Help Understand the World
All students will benefit from a basic understanding of what makes a correct logical argument and from the ability to use basic tools to study data. Students can only apply this understanding if they have no fear of mathematics and realize sometimes using multiple tools can improve perspective on a problem.

MORE MODELING
Students should recognize various growth and decay models such as linear, exponential, power and logistic. Students should be able to decide if the model is a good fit both visually and by considering the real world situation. Students should recognize that models have limitations and should be able to explain these limitations.

Students should be able to detect whether data has a linear, exponential, or power pattern. Students should use these patterns to analyze trends and predict trends.

LOGIC AND MATHEMATICAL REASONING
Students should recognize that solving mathematical problems sometimes requires persistance and multiple approaches. Students should not fear to attempt to apply the mathematical skills they possess to model a situation, or discover a pattern in data.

Students should be able to judge the validity of logical arguments.

Students should be able to create, or interpret a Venn diagram.

Students should be able to recognize and apply various apportionment methods. This applies not only to voting, but also to methods with unequal shares, for instance if the number of votes in a shareholder meeting depends on how much stock you own, or the strength of a vote in a club depends on the years of membership.

Students should be able to explain a mathematical argument in words, organize mathematical data into tables, organize mathematical data into graphs, and understand the meaning of simple mathematical formulas and equations. References: [1] The National Numeracy Network website at http://serc.carleton.edu/nnn/ Available April 23,2009
"[This] organization offers its members a network of individuals, institutions, and corporations united by the common goal of quantitative literacy for all citizens. Through national meetings, faculty workshops, research initiatives, and information sharing, the National Numeracy Network aims to strengthen the capacity of our country in the quantitative areas of business, industry, education, and research across all disciplines." The website includes links to colleges with quantitative literacy projects, to teaching resources and to its journal:Numeracy: Advancing Education in Quantitative Literacy. ----

[2] MAA Quantitative Reasoning for College Graduates: A Complement to the Standardshttp://faculty.valpo.edu/rgillman/ql/ This site includes general overview of concensus on quantitiative literacyqs the ability to: 1.Interpret mathematical models such as formulas, graphs, tables, and schematics, and draw inferences from them. 2.Represent mathematical information symbolically, visually, numerically, and verbally. 3.Use arithmetical, algebraic, geometric and statistical methods to solve problems. 4.Estimate and check answers to mathematical problems in order to determine reasonableness, identify alternatives, and select optimal results. 5.Recognize that mathematical and statistical methods have limits. Part iii of the reportincludes a section on pedagogy. It references Schoenfeld's 5 aspects of intellectual competencies.
Appendix B has a detailed list of topics recommended for inclusion in courses designed to develop quantitative reasoning. ----

[3]AMATYC Mathematics Across the Community College Curriculum Website: Links Page: http://www.mac3.amatyc.org/links.htm Available April 23, 2009 Many of the other references I include here I found by starting at the MAC^3 site. ----

[4] Materials Produced for the National Numeracy Network under a grant for the Woodrow Wilson Foundation through the National Council of Education and the Disciplines; http://wolfweb.unr.edu/homepage/jerryj/NNN/QL.html Available: April 23, 2009 These materials include modules on voting, misleading statitistics, commonly reported data like crime statistics and other topics. These modules help emphasize that every student needs to know some mathematics to be a literate citizen. Furthermore some modules could be used in non-math classes as part of a math-across-the-curriculum initiative. ----

[5] Mathematics Across the Curriculum at Dartmouth College http://www.math.dartmouth.edu/~matc/indexAvailable: April 23, 2009 The electronic resources here are well described in a paragraph from the Evergreen College Quantitative Literacy Page:
"The Center for Mathematics & Quantitative Education at Dartmouth features a collection of materials suitable for teaching quantitative literacy across all disciplines and levels. The materials on this site feature context driven mathematics and are sorted according to level in the K-12 "Little Bookshelf" and according to discipline in the 11-16 "Electronic Bookshelf"." ----

[6] Trinity College Quantitative Literacy Program http://www.trincoll.edu/depts/mcenter/mcql.htm Available April 28, 2009 This website explains how Trinity College adminsiters its Quantitiative Literacy (QL) program. It is a great source of ideas for those considering a QL program. In its course descriptions you get a sense of how accessible, practical applications can develop QL. ----

[7] Quantatative Resaoning Across the Curriculum
Brooklyn College of the City Univiesity of New York http://academic.brooklyn.cuny.edu/quant/ Available April 29, 2009 This website summarizes the results of an NSF grant from 1999 to 2001. The grant sought to "to foster the teaching of quantitative reasoning skills across the college's entire Core Curriculum." ----

[8] AMATYC,Beyond Crossroads,http://www.beyondcrossroads.com/ Available May 5, 2009
Chapter 4: Student Learning and the Learning Environment addresses math anxiety.
Chapter 6: Curricular and Program Development addresses quantitative literacy and developmental mathematics.

Specifically with regard to quantitative literacy: "Students in all college programs will be expected to: ·exhibit perseverance, ability, and confidence to use mathematics to solve problems ·perform mental arithmetic and use proportional reasoning ·estimate and check answers to problems and determine the reasonableness of results ·use geometric concepts and representations in solving problems ·collect, organize, analyze data, and interpret various representations of data, including graphs and tables ·use a variety of problem-solving strategies and exhibit logical thinking ·use basic descriptive statistics ·utilize linear, exponential, and other non-linear models as appropriate ·communicate findings both in writing and orally using appropriate mathematical language and symbolism with supporting data and graphs ·work effectively with others to solve problems ·demonstrate an understanding and an appreciation of the positive role of mathematics in their lives."

Specifcally with regard to developmental mathematics:
"Students should be able to approach mathematics through contextual, concrete, and abstract situations; apply mathematical skills to solve problems; and be able to transfer their knowledge to new situations. Students should experience multi-step problems and be comfortable working in groups and doing collaborative projects. They also should have successful experiences using technology, including calculators, spreadsheets, and other computer software, as a tool to collect, organize, and analyze data, as well as to recognize numerical and graphical patterns."

[9] NCTM Illuminations website. http://illuminations.nctm.org/Standards.aspx Available May 11, 2009 This includes links to the NCTM standards and to examples of how these standards can be taught. The list of topics above intersects all the broadest pre-K-12 standards except coordinate geometry. We may want to use language similar to what NCTM uses if we decide the list above is too specific.

Students need a foundation in mathematics to support their academic and workplace success. Quantitative literacy is required because the skills of interpreting data and reasoning in general are needed to deeply understand the arguments and information presented in today's world. Workplace mathematics will help students in broad groups of employment situations as well as in classes that apply to specific fields. Finally students who develop a mathematical habit of mind will regularly apply mathematics to find patterns and develop conjectures.

(Habit of mind? confusing...habit of thought, perhaps)To study data and develop conjectures, students will need to overcome their anxiety about mathematics and challenge themselves to use the mathematical tools they have learned.Quantitative LiteracyQuantitative literacy includes mental and technology aided computational skills and knowing when computation is appropriate, knowing when results are reasonable, as well as thinking about the world in terms of numbers and models. On a more advanced level quantitative literacy can include making and supporting arguments using numbers and mathematics.

Some core topics in quantitative literacy are listed in bold below. Sample topics within that area are given to help clarify what the core topic includes.

NUMBER SENSEStudents should recognize the value of small numbers, including significance and precision.

Students should be able to estimate products, sums, differences, and quotients of two numbers to determine the magnitude of the answer and to check the reasonableness of a computation.

Students should recognize how much precision a number has or needs to have. Students should recognize when rounded numbers are being used in contexts, such as reports in the media. (There are 110,000 people below the poverty line in the city, for instance.)

Use of Technology: Students should understand how calculators and spreadsheets use order of operations, should understand the significance of calculator results, and how to communicate these results within the precision of the problem.

FINANCEStudents should understand compound interest and simple interest, and the difference between them, as well as the effect that interest has on credit card debt.

Students should understand the nonlinear nature of compounded interest rates such as the future value of money.

Students should understand formulae that can be used to compute payments and know how to use technology such as spreadsheets to create amortization schedules.

Students should recognize the absolute size of large numbers like millions, billions and trillions and their relative size to one another.

STATISTICSStudents should understand how probability and expected value can be used to evaluate policies and to make decisions about what policies to make.

Students should be able read and make decisions based upon data from line graphs, bar graphs, histograms, box and whisker plots, scatterplots and pie charts.

Students should recognize when graph scales can lead to misinterpretation of data, and how to correctly construct graph scales for basic situations.

Students should recognize the differences between measures of central tendency, which measure is most appropriate to use and how to use multiple measures for better communication.

Students should understand percentile rank.

Students should know reasons why statistical arguments can be faulty such as those taken from [2]

including especially these

common sources of error: sampling error; misinterpreting averages or probabilitiesRATIONAL NUMBERSStudents should recognize fraction, decimal, and percent equivalencies for common fractions like halves, fourths, fifths and tenths.

Students should be able to make mental calculations involving common percentages and fractions; like 10% of a number, 1/2, 1/3 and 1/4 of a number, 110% of a number, 15% - 20% tips.

Students should be able to calculate a percent increase, or percent decrease.

Students should know the relation between base and percent. For instance the state of Illinois has more Spanish-speakers at home than New Mexico despite over 25% of New Mexicans having Spanish as a first language because of Illinois's overall population.

Students should be able to graph y = 1/x and verify points on the curve.

RATE AND PROPORTIONStudents should be able to compute and apply the average rate of change from tabular data and include the appropriate units.

Students should be able to make calculations such as those found in utility bills.

Students should be able to estimate a total given a unit rate and the number of units.

Students should know when it is more appropriate to use the actual number and when it is more appropriate to use a rate. For instance when comparing crime statistics between a medium sized and a large city the rate per 10,000 is probably more useful than the number of crimes.

MEASUREMENTStudents should know when to use length, area and volume, and understand the units involved with each.

Students should be able to explain the units used in rates and make conversions between units.

Students should be able to estimate lengths and volumes in US Standard and metric units and use these in calculation.

Students should recognize basic two- and three-dimensional shapes involving rectangular, triangular, spherical and conical components.

ALGEBRAStudents should be able to use variables as placeholders as in formulas.

Students should know what a solution to an equation is.

Students should understand scientific notation.

Students should recognize the difference between discrete and continuous functions, and understand when each type is more appropriate.

Students should be able to judge when to use an algorithm and be able to apply the steps sequentially.

? Students should be competent at problem solving ... writing algebraic statements for descriptions of problems

? Students should be able to evaluate expressions using the order of operations, including exponents and parentheses

? Students should understand the difference between a variable and a constant

Mathematics in the WorkplaceIn addition to general quantitative literacy, many fields of study require an understanding of statistics and modeling; and require enough algebra to understand, use and create spreadsheets.

ADDITIONAL STATISTICS SPECIFIC TO THE WORKPLACEStudents should be able to read data presented in a table and write data into a table. Students should recognize that matrices (arrays) can be used to manipulate data as well as store data.

Students should have a basic understanding of reading graphs and tables as detailed in the Quantitative Literacy section above.

MODELINGStudents should understand linear, exponential, power and logistic growth as described in the Math as a Tool to Understand the World Section below.

Students should be able to design an experiment that collects appropriate data in order to find a reasonable model and then use that model to make predictions and/or observations.

Students should understand the meaning of variable, including formulas and evaluation. Students should be able to use spreadsheet cell references as variables.

Students should be able to use spreadhshhets to investigate scenarios.

Math as a Tool to Help Understand the WorldAll students will benefit from a basic understanding of what makes a correct logical argument and from the ability to use basic tools to study data. Students can only apply this understanding if they have no fear of mathematics and realize sometimes using multiple tools can improve perspective on a problem.

MORE MODELINGStudents should recognize various growth and decay models such as linear, exponential, power and logistic. Students should be able to decide if the model is a good fit both visually and by considering the real world situation. Students should recognize that models have limitations and should be able to explain these limitations.

Students should be able to detect whether data has a linear, exponential, or power pattern. Students should use these patterns to analyze trends and predict trends.

LOGIC AND MATHEMATICAL REASONINGStudents should recognize that solving mathematical problems sometimes requires persistance and multiple approaches. Students should not fear to attempt to apply the mathematical skills they possess to model a situation, or discover a pattern in data.

Students should be able to judge the validity of logical arguments.

Students should be able to create, or interpret a Venn diagram.

Students should be able to recognize and apply various apportionment methods. This applies not only to voting, but also to methods with unequal shares, for instance if the number of votes in a shareholder meeting depends on how much stock you own, or the strength of a vote in a club depends on the years of membership.

Students should be able to explain a mathematical argument in words, organize mathematical data into tables, organize mathematical data into graphs, and understand the meaning of simple mathematical formulas and equations.

References:[1]

The National Numeracy Networkwebsite at http://serc.carleton.edu/nnn/ Available April 23,2009"[This] organization offers its members a network of individuals, institutions, and corporations united by the common goal of quantitative literacy for all citizens. Through national meetings, faculty workshops, research initiatives, and information sharing, the National Numeracy Network aims to strengthen the capacity of our country in the quantitative areas of business, industry, education, and research across all disciplines."

The website includes links to colleges with quantitative literacy projects, to teaching resources and to its journal:Numeracy: Advancing Education in Quantitative Literacy.----

[2]

MAA Quantitative Reasoning for College Graduates: A Complement to the Standardshttp://faculty.valpo.edu/rgillman/ql/This site includes general overview of concensus on quantitiative literacyqs the ability to:1.

Interpret mathematical models such as formulas, graphs, tables, and schematics, and draw inferences from them.2.

Represent mathematical information symbolically, visually, numerically, and verbally.3.

Use arithmetical, algebraic, geometric and statistical methods to solve problems.4.

Estimate and check answers to mathematical problems in order to determine reasonableness, identify alternatives, and select optimal results.5.

Recognize that mathematical and statistical methods have limits.Part iii of the reportincludes a section on pedagogy. It references Schoenfeld's 5 aspects of intellectual competencies.Appendix B has a detailed list of topics recommended for inclusion in courses designed to develop quantitative reasoning.

----

[3]

AMATYC Mathematics Across the Community College Curriculum Website: Links Page: http://www.mac3.amatyc.org/links.htm Available April 23, 2009Many of the other references I include here I found by starting at the MAC^3 site.----

[4] Materials Produced for the National Numeracy Network under a grant for the Woodrow Wilson Foundation through the National Council of Education and the Disciplines; http://wolfweb.unr.edu/homepage/jerryj/NNN/QL.html Available: April 23, 2009These materials include modules on voting, misleading statitistics, commonly reported data like crime statistics and other topics. These modules help emphasize that every student needs to know some mathematics to be a literate citizen. Furthermore some modules could be used in non-math classes as part of a math-across-the-curriculum initiative.----

[5] Mathematics Across the Curriculum at Dartmouth Collegehttp://www.math.dartmouth.edu/~matc/indexAvailable: April 23, 2009

The electronic resources here are well described in a paragraph from the Evergreen College Quantitative Literacy Page:"The Center for Mathematics & Quantitative Education at Dartmouth features a collection of materials suitable for teaching quantitative literacy across all disciplines and levels. The materials on this site feature context driven mathematics and are sorted according to level in the K-12 "Little Bookshelf" and according to discipline in the 11-16 "Electronic Bookshelf"."

----

[6] Trinity College Quantitative Literacy Programhttp://www.trincoll.edu/depts/mcenter/mcql.htm Available April 28, 2009

This website explains how Trinity College adminsiters its Quantitiative Literacy (QL) program. It is a great source of ideas for those considering a QL program. In its course descriptions you get a sense of how accessible, practical applications can develop QL.----

[7] Quantatative Resaoning Across the CurriculumBrooklyn College of the City Univiesity of New York

http://academic.brooklyn.cuny.edu/quant/ Available April 29, 2009

This website summarizes the results of an NSF grant from 1999 to 2001. The grant sought to "to foster the teaching of quantitative reasoning skills across the college's entire Core Curriculum."----

[8] AMATYC,Beyond Crossroads,http://www.beyondcrossroads.com/ Available May 5, 2009Chapter 4: Student Learning and the Learning Environment

addresses math anxiety.Chapter 6: Curricular and Program Development

addresses quantitative literacy and developmental mathematics.Specifically with regard to quantitative literacy::"Students in all college programs will be expected to· exhibit perseverance, ability, and confidence to use mathematics to solve problems

· perform mental arithmetic and use proportional reasoning

· estimate and check answers to problems and determine the reasonableness of results

· use geometric concepts and representations in solving problems

· collect, organize, analyze data, and interpret various representations of data, including graphs and tables

· use a variety of problem-solving strategies and exhibit logical thinking

· use basic descriptive statistics

· utilize linear, exponential, and other non-linear models as appropriate

· communicate findings both in writing and orally using appropriate mathematical language and symbolism with supporting data and graphs

· work effectively with others to solve problems

· demonstrate an understanding and an appreciation of the positive role of mathematics in their lives."

Specifcally with regard to developmental mathematics:"Students should be able to approach mathematics through contextual, concrete, and abstract situations; apply mathematical skills to solve problems; and be able to transfer their knowledge to new situations. Students should experience multi-step problems and be comfortable working in groups and doing collaborative projects. They also should have successful experiences using technology, including calculators, spreadsheets, and other computer software, as a tool to collect, organize, and analyze data, as well as to recognize numerical and graphical patterns."

[9]NCTM Illuminations website. http://illuminations.nctm.org/Standards.aspx Available May 11, 2009This includes links to the NCTM standards and to examples of how these standards can be taught. The list of topics above intersects all the broadest pre-K-12 standards except coordinate geometry. We may want to use language similar to what NCTM uses if we decide the list above is too specific.