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**(1) Carleton College** (Carleton College 2009)

*What must a course include to satisfy the QRE requirement?*

Students are required to have three "Quantitative Reasoning Encounters" (QRE). The goal of the requirement is to increase students' appreciation for the power of QR and to enhance their ability to evaluate, construct, and communicate arguments using quantitative information. A course designated as a "Quantitative Reasoning Encounter" will include at least one substantial assignment or module designed to enhance one or more of the following QR skills:

- Possessing the habit of mind to consider what numerical evidence might add to the analysis of a problem;
- Identifying appropriate quantitative or numerical evidence to address a question;
- Locating or collecting data;
- Interpreting numerical evidence properly including recognizing the limitations of methods and sources used;
- Effectively communicating arguments that involve numerical or quantitative evidence.

*What constitutes a "substantial assignment or module"?*

An assignment that comprised a significant proportion of a student's grade for the class and involved extensive research and/or writing would obviously be substantial. However, these need not be the only criteria for such designation. An assignment that required sustained attention or effort from students (ex: regular news reading), was central to the main narrative of the course, or represented a significant allocation of in-class activity (ex. simulation or lab experience) might also be considered substantial even if these elements were not graded or were evaluated on a simplified scale (pass/fail). Faculty members are the final arbiters of what constitutes a substantial QRE module or assignment.

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**(2) DePauw University**

Quantitative Reasoning is one of three competence requirements (also included are expository writing and oral communication). Courses designated as fulfilling the quantitative reasoning (Q) competency requirement encourage:

- understanding quantitative concepts, representational formats and methodologies of a particular discipline;
- evaluating quantitative evidence and arguments;
- making decisions based upon quantitative information; and
- learning through problem-solving, laboratory experiments and projects.

Students must achieve Q certification by the end of their junior year. If students do not achieve certification by the end of the first semester of the junior year, they must complete a Q course each succeeding semester until certification is achieved. Q courses are offered in several academic subjects each semester, subject to approval by the Q committee, and normally carry one course credit each. They emphasize both quantitative reasoning and mastery of course content.

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**(3) Hollins University** (Diefenderfer, Doan, and Salowey 2006: 42-43)

At Hollins University, students must fulfill a QR Basic Skills requirement (either pass a test or enroll in a course entitled "Introduction to Quantitative Reasoning"). The goals of this course are (1) to understand mathematical and statistical reasoning and (2) to use appropriate mathematical and/or statistical tools in summarizing data, making predictions, and establishing cause-and-effect relationships. The QR applied skills requirements can be satisfied by passing a course designated as a QR applied course. The goals of the QR applied courses are to:

(a) give students the opportunity to apply mathematical and statistical reasoning in a chosen discipline, and

(b) to involve students in the application of quantitative skills to problems that arise naturally in the discipline in a way that advances the goals of the course is not merely a rote application of a procedure.

*Defining QR Across the Curriculum*

Each QR applied course must include at least two QR projects. A project might, for example, include data collection, discussion of the data, collaborative work on finding appropriate uses of the data, and use of appropriate technology in presentation and writing. The end results of each QR project should be a written assignment that includes a statement of the problem, an explanation of the methods used, and a summary of the results. When appropriate, the written assignment should discuss any limitations encountered and possible improvements to the procedure and/or results

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**(4) Kalamazoo College** (Fink and Nordmoe 2006)

The Quantitative Reasoning Committee *recommended* a Quantitative Reasoning Program "across the curriculum" to "provide students with repeated practice and coaching in a variety of environments, help them strengthen their analytical abilities, and cultivate a more confident understanding of the meanings (and uses) of numbers and their presentation" (Fink and Normoe 2006: 51).

*Defining QR Across the Curriculum*

A course will satisfy the Quantitative Reasoning requirements if students in the course develop their ability in all of the first three areas and three of the last four following areas:

(1) **Organize Ideas Effectively**. This includes using quantitative language and mathematical symbols to clarify ideas and using sketches, diagrams, graphs, tables, and other mathematical models to analyze situations.

(2) **Communicate Ideas and Information Clearly**. This includes writing and speaking about quantitative ideas clearly in words, using mathematical notation correctly, presenting relevant data effectively, and using graphs and tables when appropriate

(3) **Construct and Defend an Argument Using Evidence Persuasively**. This includes reasoning deductively, using statistics appropriately, and using estimates and error analysis correctly

(4) **Interpret and Create Graphs and Tables**. This includes understanding Cartesian coordinates, converting information given by a formula, graph, or table to another format, understanding the significance of the slope and concavity of a graph, and understanding histograms and others ways to present data. It also includes using a software package such as Excel, CricketGraph, DeriveMaple, Mathematica, or MATLAB, to work with data presentation

(5) **Use Various Measurement Scales to Interpret Data**. This includes converting between scales, converting units, and using appropriate scaling in graphs and tables.

(6) **Apply Simple Mathematical Models**. This includes using various functions as mathematical models, and using algebra to make predictions from mathematical models.

(7) **Interpret Statistics**. This includes the understanding of mean, standard deviation, correlation coefficient, and regression lines, and using software packages such as Fastat, Minitab, and SPSS.

For many students, the requirement was satisfied through a required course in the major (as a result of this requirement, a new course was also created in Sociology and Anthropology). But for some, such as English majors, there was no natural way to satisfy the requirement so the Department of Mathematics added a new course called "Quantitative Reasoning and Statistical Analysis."

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**(5) Lawrence University** (Haines and Jordan: 2006)

Courses satisfying the requirement in Mathematical Reasoning or Quantitative Analysis would be expected to meet the following criteria^{1}:

1. In such courses, at least 50% of the final grade must be based on evaluated quantitative exercises.

2. Such courses must require students to do a substantial amount of quantitative or mathematical work distributed over the course of the term.

3. Courses satisfying this requirement must provide explicit instruction in quantitative methods and quantitative reasoning. Exercises in these courses might take the form of

- Symbolic proofs (either logical or mathematical)
- Statistical or graphical analysis of numerical data
- Problem solving using mathematical methods
- Implementation of computer algorithms

All students are required to take at least one of these classes.

Faculty who are interested in submitting a course to fulfill the Mathematical Reasoning/Quantitative Analysis Requirement at Lawrence University must submit the attached form 1337199033 (Acrobat (PDF) 15kB May16 12).

^{1}http://www.lawrence.edu/mfhe/www_core/Everyone/quantitativecourseform.pdf

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**(6) Macalester College** (Bressoud 2009)

Quantitative thinking is defined to include:

1. **Describing the World Quantitatively:** Much of quantitative thinking involves quantitative or statistical descriptions of social and natural phenomena. This includes descriptions of patterns and variations and rates of change, such as linear or exponential growth. Understanding descriptive statistics and the various modes of presentation of quantitative data is central. Students should be able to distinguish when quantitative approaches are appropriate and when they are not.

2. **Evaluating Sources and Quality of Data:** Students of quantitative thinking should also understand the sources of data, including the processes of collecting or producing data. This may involve understanding how to assess the reliability and validity of measurements and elements of probability and sampling, including sources of bias and error.

3. **Association and Causation:** The quantitative thinker knows the ways that associations between factors are established by observation, experiment or quasi-experiment. It is important to be able to establish the meaning of an association or correlation and learn the protocols for weighing the statistical significance and theoretical importance of findings, including inferring causation.

4. **Trade-Offs**: Most decisions, whether public or private, individual or societal, may be thought of as involving conflicting goals. Much of the debate on public issues involves disagreement about the value of the different goals. Where there are conflicting goals, quantitative thinking offers techniques for weighing the relative impact of policy options. While there rarely is a single correct outcome in the face of such conflicts, the quantitative thinkers can bring measure and balance to policy discussion.

5. **Uncertainty and Risk:** Few things in life are certain; decisions and debate often revolve around unknowns. The quantitative thinker possesses skills that can be used to assess, compare and balance risks, and understands the limits and strengths of these techniques. The quantitative thinker knows that, in the face of the unknown, if not the unknowable, we often rely on conditional statements and probabilities in making decisions and can evaluate conclusions drawn from conditional statements.

6. **Estimation, Modeling, and Scale:** The quantitative thinker understands that quantities vary over huge ranges; 'big' and 'small' are not absolute notions but depend on context or scale. Quantitative thinkers appreciate the value and limitations of abstracting out detailâ€”constructing modelsâ€”and that the sensitivity of model results to assumptions can and should be reported along with the model results.

Courses in Quantitative Thinking (QT) are designated at one of three levels:

- **Q3** The great majority of material covered in a Q3 course focuses on quantitative topics, and a Q3 course covers all or nearly all of the six learning goals.

- **Q2** At least half of the material covered in a Q2 course focuses on quantitative topics, and a Q2 course covers the majority of the six learning goals.

- **Q1** A Q1 course covers some of the six learning goals, and quantitative thinking elements represent some of the overall material covered in the course.

A student satisfies the QT requirement by taking either one Q3 course, two Q courses at least one of which is Q2, or any three Q1 courses. While the student who opts for three Q1 courses may not experience all six of the core goals of QT, it was decided that trying to keep track of which had not been covered would be a logistical nightmare. In view of the extended exposure to QT such students would receive, we could live with less than perfect coverage.

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**(7)** **Stockton College**

As described on their webpage, "before graduating, all matriculated students must complete three quantitative-reasoning-designated courses, including at least one Q1 (quantitative-reasoning-intensive) course and at least one Q2 (quantitative-reasoning-across-the-disciplines) course. A Q1 course must be completed during the first year.

Transfer students are also subject to the quantitative reasoning requirement. Up to two transfer courses in mathematics and statistics may be credited as Q1 courses and counted toward the requirement. All Q2 courses must be completed at Stockton.

Q-designated courses that carry fewer than 4 credits or transfer courses that carry fewer than 3 credits do not count toward meeting the quantitative reasoning requirement.

This requirement specifies the minimum number of quantitative-reasoning-designated courses needed for graduation. To facilitate their quantitative development, students are encouraged to take as many of these courses as possible throughout their undergraduate curriculum. . .

Stockton offers two types of quantitative-reasoning-designated courses: Quantitative-Reasoning-Intensive (Q1) and Quantitative-Reasoning-Across-The-Disciplines (Q2) courses. This designation indicates the role and function of quantitative reasoning in the course, not the degree of difficulty. Q-designated courses appear throughout the curriculum, in Program and General Studies courses.

Q1 and Q2 courses emphasize mathematical problem solving with special attention given to the development of problem-solving approaches. In addition, these courses stress the importance of the communication of mathematical ideas in both written and oral forms.

Q1 - Quantitative-Reasoning-Intensive Courses: Mathematical thinking is the primary focus of study. Q1 courses emphasize the mathematical structures underlying various phenomena. Although focused on mathematical reasoning, Q1 courses provide ample opportunities for investigating diverse applications of the concepts discussed. These courses draw rich connections among different areas of mathematics. In a Q1 course, the majority of class time is spent on mathematical concepts and procedures. Students work on mathematics during virtually every class session. The quality of their mathematical work is the major criterion for evaluating student performance in the course. Examples of Q1 courses are MATH 2215 Calculus I; FRST 2310 Algebraic Problem Solving; and CSIS 1206 Statistics.

Q2 - Quantitative-Reasoning-Across-the-Disciplines: In a Q2 course, the focus is on disciplinary or interdisciplinary content outside of mathematics. Quantitative reasoning is used as a tool for understanding this content. Q2 courses feature applications that use real-world data and situations; applying a quantitative perspective to the concepts in the course results in a fuller understanding of both the disciplinary concepts and the mathematical concepts. In a Q2 course, at least 20 percent of class time involves quantitative reasoning. Students are expected to demonstrate their ability to apply mathematical ideas to the course content. Both mastery of disciplinary content and quantitative proficiency are used to evaluate student performance. Examples of Q2 courses include ARTV 2121 Black and White Photography, PSYC 3242 Experimental Psychology, GNM 2182 Atom, Man, Universe; and CHEM 2110 Chemistry I General Principals. Unless a course is designated "intrinsic", each individual instructor has the option to apply for Q2 designation through the QUAD central task force.

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**(8) Wellesley College** (Taylor 2006)

At Wellesley College, students must satisfy a QR "basic skills" component (by passing a QR assessment given upon entering the College or by passing the QR basic skills course). In addition, students must pass a "QR overlay course" to satisfy the second part of the QR requirement. "The QR overlay courses are intended to teach students how numerical data are analyzed and interpreted in various disciplines. The term 'overlay' indicates students may use one of the designated QR courses to simultaneously satisfy another of the College's distribution requirements. . . . Guidelines for the QR overlap classes are as follows":

QR Overlap Classes need to Balance the Following Five Objectives:

a) **Literacy**. The number of topics and the depth of coverage should be sufficient to ensure that students have the basic knowledge they need in order to function in real-life situations involving quantitative data.

b) **Authenticity**. Students should have experience in using authentic numerical data. The experience should arise naturally in the context of the course and actually advance the work of the course. Only with such experience is the literacy goal likely to be realized.

c) **Applicability**. The examples used in an overlay class should be adequate to convince the average student that the methods used in the analysis of data are of general applicability and usefulness

d) **Understanding**. A student's experience with data analysis should not be limited to rote application of some involved statistical procedure. Rather, students should understand enough of what they are doing so that their experience of data analysis is likely to stay with them, at least as a residue of judgment and willingness to enter into similar data analyses in the future.

e) **Practicality**. The breadth of topics covered and the depth of coverage should be consistent with what an average Wellesley student can realistically absorb in a course that devotes only a part of its time to data analysis.

*Minimum Exposure to the Analysis of Data*

The following topics need to be addressed in a QR overlay course in order to satisfy the objectives of literacy and understanding.

a) **Framework for Data Analysis**. A QR overlay course should provide an overview of how empirical questions or hypotheses can be raised, how relevant data can be collected and analyzed to address these questions, and finally, what conclusions these data allow. Students should formulate questions that arise in the context of the course and that can be answered by analyzing data. They should then decide what type of data to collect, how to collect and analyze these data, and what conclusions these data support.

b) **Collecting Data**. A QR overlay should address issues of data quality. Are the data representative or biased? Are the data that are collected really relevant to the question being investigated? Certain courses might stress the importance of random samples, experiments versus observation studies, blind versus double blind experiments, and so forth.

c) **Representing Data**. A QR overlay course should stress different methods of representing data, including numerical representations (tables of data), visual representations (pie charts, scatter-lots, line graphs, and histograms), as well as verbal representations (writing reasonable captions that describe a graph).

d) **Summarizing Data**. Students need to be exposed to different ways of summarizing data, including verbal summaries of data sets. They also need to study different measures of central tendency (including the mean, median and mode) as well as different measures of dispersion (including the range, standard deviation, percentile ranks, index of diversity, and index of qualitative variation).

e) **Probability**. Because of the random component in sampling from a population, students should have some understanding of basic probability. This must include a working knowledge of how and when to use the addition and multiplication rules, the concept of conditional probability, and the vocabulary of statistical independence and mutual exclusivity.

f) **Distribution**. Students in a QR overlay course should work with examples of different distributions, including normally distributed data and various non-normally distributed data. They should know when to expect that a population will be normally distributed, and what it means for a distribution to be skewed. They should know what the mean, median and standard deviation tell about a distribution. Finally, they should know that a normal distribution is fully specified by its mean and standard deviation and that the percentage of the population on a given interval can be determined from a table or a formula.

g) **Sampling**. A QR overlay course should discuss sampling and stress the distinction between "sampling" and "collecting data." It should introduce the notion of sample mean, and discuss why the sample mean might well vary from the population mean. The distinction between sample statistics (e.g., the mean of a sample) and population parameters (e.g., the mean of the population) should be emphasized.

*Authentic Application*

At least one of the following applications should be addressed to satisfy the objectives of authenticity and applicability:

a) **Issues Regarding Sampling**. An overlay course could address problems that can arise when one attempts to ascertain certain characteristics of a population by testing a sample of that population. Such issues include how one obtains a random sample and how one detects sample biases.

b) **Making and Justifying Inferences from Data**. A QR overlay course could discuss confidence intervals and hypothesis tests. It could explore how one determines whether a measured variation cannot plausibly be attributed to chance alone

c) **Regression Analysis**. A QR overlay course could study various methods for fitting curves to data and for analyzing the deviations of the data from these curves.

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**(8) University of Texas at San Antonio**

The student learning goals of the QEP will help undergraduate students (a) acquire basic quantitative literacy and numeracy skills, (b) effectively communicate the results of their quantitative analysis, and (c) acquire discipline-specific advanced quantitative skills.

The student learning goals of the Quality Enhancement Plan (QEP) are designed to "help undergraduate students (a) acquire basic quantitative literacy and numeracy skills, (b) effectively communicate the results of their quantitative analysis, and (c) acquire discipline-specific advanced quantitative skills. . . Courses that have been redesigned by integrating quantitative literacy will be designated as Q-courses, which enable students to develop their data-reasoning skills. Each academic year additional courses are selected to participate in the QL program."

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**Sources Cited**

Bressoud, David. 2009. "Establishing the Quantitative Thinking Program at Macalester." *Numeracy* 2(1): Article 3.

Carleton College. 2009. "Graduation Requirements." Dean of the College Web Site.

Diefenderfer, Caren, Ruth Doan, and Christina Salowey. 2006. "The Quantitative Reasoning Program at Hollins University." In *Current Practices in Quantitative Literacy*, edited by Rick Gillman. Washington, DC: Mathematical Association of America. Pp. 41-48.

Fink, John B. and Eric D. Nordmoe. 2006. '"A Decade of Quantitative Reasoning at Kalamazoo College." In *Current Practices in Quantitative Literacy*, edited by Rick Gillman. Washington, DC: Mathematical Association of America. Pp. 51-62.

Haines, Beth and Joy Jordan. 2006. "Quantitative Reasoning Across the Curriculum." In *Current Practices in Quantitative Literacy*, edited by Rick Gillman. Washington, DC: Mathematical Association of America. Pp. 63-68.

Taylor, Corrine. 2006. "Quantitative Reasoning at Wellesley College." In *Current Practices in Quantitative Literacy*, edited by Rick Gillman. Washington, DC: Mathematical Association of America.