Guidelines for Quantitative Reasoning

Introduction

            The requirements for general education mandate that each student take a minimum of one course in quantitative reasoning.  The importance of the quantitative reasoning requirement can’t be understated.  This one course has to satisfy the requirement that the Organization of Mid Atlantic States calls one of its “essential skills” of general education, not to mention the expectations of parents, employers and taxpayers.  The provost declares that quantitative problem solving is one of the tools we want to foster in our students.  Since the transformative process calls for more intensive courses that promote deep understandings, this is not the time to weaken standards. 

 

            The question “what is quantitative reasoning” immediately arises.  The Dean’s Response to GEAC’s Guide for Liberal Learning contains the following, a mere five sentences, defining the requirement. 

 

Quantitative Reasoning Outcomes
Apply math to society	Students will understand how real-world problems and social issues can be analyzed using the power and rigor of mathematical and statistical models.
Understand math representations	Students will be able to evaluate representation and inferences that are based on quantitative information.
Interpret math models	Students will be able to interpret mathematical models such as formulas, graphs, and tables, and draw inferences from them.
Find math answers	Students will be able to estimate and check answers to mathematical problems in order to determine reasonableness, identify alternatives, and select optimal results
Use math methods	Students will be able to use arithmetical, algebraic, geometric, and statistical methods to solve problems, but learn to recognize the limitations of mathematics and statistics as well.

 

            The following document is an interpretation of the above guidelines in an effort to clarify what is and what is not quantitative reasoning.  It is not the intent of this document to crush innovative ideas.  This should serve primarily as a point of comparison.  We accept that courses that do not conform to the following guidelines may satisfy the requirement; however, we feel it is incumbent on the author of such a course to justify, in a convincing argument, such deviations.

Quantitative Reasoning and Mathematics

            Each heading in the Dean’s Response contains the word math.  Each sentence is filled with mathematical terms like formulas, algebraic, rigor, and statistics.  GEAC’s Design for Liberal Learning insists that the quantitative reasoning requirement is only satisfied by a minimum of one course with goals “aligned with those published by the Mathematics Association of America”.  Clearly, quantitative reasoning courses are courses in mathematics.  While it is understood that such courses are not intended to be traditional math fare, or even taught by mathematicians, the tools and methods of reasoning must be primarily mathematical.  No one would classify a course containing a few anecdotes, as a history course.  Similarly a course containing some mathematics is not a quantitative reasoning course.

Quantitative Reasoning Outside of Mathematics

            Many students are turned off by the traditional approach to mathematics.  These students are not likely to be successful by continuing along the same path.  One goal of quantitative reasoning is to combine mathematics with a subject that engages their interest.  It is hoped that these students will gain an appreciation of math and “will understand how real-world problems and social issues can be analyzed using the power and rigor of mathematical and statistical models.” 

            Most math courses traditionally contain applications structured around a math topic.   Quantitative reasoning opens the door to courses focused on a non-mathematical subject, treated with a wide array of quantitative tools.  Such courses work best as quantitative reasoning when the subject matter and mathematics are seamlessly combined.

The Level of Mathematics

            The Dean’s Response calls for problems to be “analyzed using the power and rigor of mathematical and statistical models.”  Rigor means strict mathematical reasoning, rarely seen at the high school level.  This implies that the depth of mathematics in these courses is anything but elementary.  The Dean’s Response also calls for high ordered reasoning.  Students are expected to “evaluate representations”, “interpret mathematical models”, “determine reasonableness” and “recognize limitations”.  This implies that the students need a deep understanding of the methods they employ.  Specifically they need to comprehend the meaning of the calculations and answers and determine if such methods are appropriate.  Plugging numbers into equations or a calculator will clearly not suffice. 

The Breadth of Mathematics

            The Dean’s Response calls for students to master “arithmetical, algebraic, geometric, and statistical methods”.  A literal reading of this requirement suggests that every element must be addressed; however not every subject lends itself to all mathematical tools.  Breadth within a mathematical subject may be a substitute for breadth between such subjects.  For example, if the mathematical content is primarily statistics, a variety of statistical methods should be employed throughout the course.

The Use of Technology

            Modern calculators and computers, if used properly, can greatly enhance a quantitative reasoning course.  Traditional mathematics courses use a substantial amount of time teaching hand calculation.  Technology can now graph equations, compute derivatives, perform matrix operations and analyze a data set statistically.  This opens up weeks of time to be devoted to other aspects of a course.

            It must be acknowledged that technology is only a tool.  Using a hammer to break a piggy bank is not woodworking.   Similarly typing numbers into a calculator or a computer is not quantitative reasoning.  The nature of skills is not determined by tools but rather how they are used.  The high level reasoning required by the Deans’ Response can only be achieved if students understand what the values they enter mean, what the technology does with these values, how to interpret the output and why the procedure is appropriate to the given problem. 

            The time constraint of a semester and the goals of a course are frequently at odds.  It is not expected that every method be explained in excruciating detail.  Occasionally technology may be used to save time; however, the time saved should be used to emphasize other aspects of quantitative reasoning.  Technology, if used, should enhance mathematics, not bypass it.

Creating a Quantitative Reasoning Course

            Clearly it’s impossible to describe every format for a quantitative reasoning course.  This section hopes to merely provide a few examples of how this might be accomplished.  Since a quantitative course requires breadth, one or even a few of the following will not suffice. 

 

Modeling with equations:      Given a model with either an intuitive or formal description:

• Translate intuitive descriptions into variables connected by equations
• Solve the equations of a model by hand or with an equation solver.
• Use a program to graph the variables of a model.  Given one value, can you estimate another?  How can you tell if there is a solution or a unique solution?
• Model the equations in a program or on a spreadsheet.  How is a change in one value reflected in the change of the others?  What impact does this have on measurement errors?

Statistical Analysis:   After entering data into a statistical program

• Change all the values systematically, like adding one or doubling each.  How does this effect the program’s output.  Use this to give an intuitive meaning to standard deviations and means.
• A theorem states that roughly two thirds of the data should be within one standard deviation of the average for “normal” curves.  Have the students determine how accurate this is.  Discuss what types of data tend to have normal distributions and what do their distribution curves look like.  Have them create a dataset that is visually not normal and test the theorem again.
• Have the students calculate the variance and standard deviation of a very small dataset by hand.  How can variance be thought of as an average?

Regression Analysis: Given a data set apply a linear regression

• Determine visually if the line truly represents the data.
• Discuss examples of data that might be expected to have a linear distribution.  What data sets might not?  Test their guess.
• Randomly select a small sample of the data.  How does this effect the regression?

 

            Note that the above examples do more than just use mathematics.  They examine the modes of mathematics themselves.  

           

            There are many more areas that might fall under the category of quantitative reasoning such as error analysis and sampling.  In addition to the above generic topics, all subjects have their own forms of mathematics.  Music theory deals with ratios, its physics deals with waves and its technology deals with filters applying function transformations.           
The graphic arts use ray tracing, physical models and fractal generation in its computer art.  Even in literature, statistics can be applied to determine authorship of disputed documents.

Justifying Quantitative Reasoning

            The syllabus of a prospective quantitative reasoning course has to justify this classification.  This is difficult to do because of the wide range of formats and content possible in such a class.  Making vague references to key words in the Deans’ Response won’t answer questions of depth and breadth.  Telling a mathematician that a course uses functions is like telling an English professor that a course uses words.  In order to evaluate a quantitative reasoning course, it’s necessary to know something about the specific content and methods used.  On the other hand, providing a complete description of all mathematical methods and assignments would be unwieldy and time consuming. 

 

            We suggest that an appendix be added to the document which contains some typical examples of mathematical assignments in the course.  These examples should give insight into the depth of the mathematics.  This should be followed by a brief statement of how often such assignments are given, indicating the breadth of the mathematics.  In addition to the appendix, it would help if the title of course topics included a mathematical component.  So, for example, instead of The Structure of Fish Populations it might be better to have something like Statistical Analysis of Fish Populations, in order to justify the breadth component.