Quantitative Skills in the Geosciences
Cathryn A. Manduca, Carleton College, 2002
This essay argues that geoscience faculty should care about integrating quantitative skills into their teaching and synthesizes many of the seminal works on the subject available in 2002.
Jump down to: What Do We Mean by Quantitative Skills?  What is quantitative literacy?  What skills are important for all students?  What skills are important for geoscience students?  What can we learn from our sister disciplines?  How will we know when we are effective?
Why are quantitative skills an important part of geoscience education?
Ranging from the ability to interpret numerical data to the development of theoretical and computational models, quantitative skills are a fundamental part of a geoscientist's toolbox. Similarly, as data access, computer models, statistics, and numbers become more important in everyday life, all students should be well equipped to make decisions based on quantitative data. The Earth Sciences play an important role in teaching quantitative skills at the undergraduate level. Earth science courses prepare future geoscientists, future teachers, and a wide variety of professionals in other fields that rely on interpretation of quantitative environmental data. In addition, a large percentage of nonscientists receive their introduction to science in entrylevel geoscience courses. Traditionally many geoscience courses have been viewed as lacking quantitative rigor. While this is clearly not true of the research enterprise, the undergraduate curriculum has often deemphasized the quantitative aspects of the science, particularly at the introductory level.
The recognition that teaching quantitative skills in the geosciences is critical to our country's educational goals for all undergraduate students (e.g. NSF, 1996 ) led to two NSFfunded national workshops on this topic: Building the Quantitative Skills of Majors and Nonmajors in Earth and Planetary Science Courses (Macdonald et al, 1998 ) and Building Quantitative Skills of Students in Geoscience Courses (Macdonald, Srogi and Stracher, 2000 (Eds.) ). These workshops and symposia at professional society meetings (Stracher, 1998 ; Macdonald, Srogi and Stracher, 1999 ) have begun to identify the work that is taking place on campuses across the country and to recognize effective approaches to teaching quantitative skills in the geosciences. Special issues of Mathematical Geology (Stracher, 2000 (Ed.) ) and Journal of Geoscience Education (Macdonald, Srogi and Stracher, 2000 (Eds.) ) present some of this work.
Faculty have a wide variety of perspectives on the role that quantitative skills should play in their courses. Often this differs from course to course, particularly between entrylevel, upper division, and graduate courses. Macdonald and Bailey in their article Integrating the Teaching of Quantitative Skills Across the Geology Curriculum in a Department (Macdonald and Bailey, 2000 ) write:
We identified quantitativeskills development as a critically important goal because many geologic problems require quantitative solutions, many graduate schools and employers expect new employees to have a significant level of quantitative proficiency, and the newer faculty are involved in research that is quantitative. In addition, introductory naturalscience courses at William and Mary that meet the new general education requirements are expected to address the role of mathematics in science, providing additional incentive to make such courses more quantitative.
Understanding the philosophical underpinnings guiding our decisions for incorporating quantitative learning in courses is an essential step in articulating why quantitative skills are an important part of training for both majors and introductory students. What role should quantitative learning play in geoscience courses? What is the rationale for this decision in courses serving nonmajors? majors? research students?
What do we mean by quantitative skills?
Geoscience faculty and mathematics faculty refer to a wide variety of concepts and learning goals as "quantitative skills". These range from the ability to read graphs and change units to skills as sophisticated as representing real world observations in numerical models. (See for example Varieties of Quantitative Literacy (more info) ).
Ido Gaal, in his essay Numeracy: Imperatives of a Forgotten Goal (Gaal, 1997 ) describes three different situations in which we use our quantitative skills:
 Computational Situationswhere a number is the desired result
 Decisionmaking Situationswhere numerical results guide our decision making
 Interpretive Situationswhere quantitative information is combined with other information and value judgments to come to an understanding
In Thinking Quantitatively about Science (Rutherford, 1997 ), James Rutherford concludes that seven fundamental understandings underpin the quantitative thinking of scientists.
 The Nature of Mathematics
 Computation and Estimation
 Numbers and Quantities
 Measurement
 Relationships
 Uncertainty
 Critical Response Skills
The American Mathematical Association of Two Year Colleges, in their document Crossroads in Mathematics: Standards for introductory college mathematics before calculus. (more info) , recognizes intellectual skills and content skills.
Content Standards
 Number Sense
 Symbolism and Algebra
 Geometry
 Function
 Discrete Mathematics
 Probability and Statistics
 Deductive Proof
Intellectual Development standards
 Problem solving
 Modeling
 Reasoning
 Connecting with Other Disciplines
 Communicating
 Using Technology
 Developing Mathematical Power
Geoscience faculty teach quantitative skills in two fundamentally different contexts. At the introductory level, we must ask the question, what quantitative skills are important for the full range of students enrolled in these courses, many of whom will have little further scientific training. A particularly important population in these courses is those students who will become teachers. In thinking about majors, it is important to consider our goals for these students and the diversity of their future career pathways. What are our learning goals for each group of students? How do we taylor our programs to meet the needs of each of these groups which are in themselves diverse?
What is quantitative literacy?
Quantitative literacy, the facility to use and understand quantitative information, is now recognized as an important outcome for both K12 and undergraduate education. To be successful citizens, students must be as comfortable with numbers as they are with words. Lynn Steen has collected essays from a wide variety of viewpoints in two books that illuminate why these skills are so fundamental:
 Why Numbers Count? Quantitative Literacy for Tomorrow's America (Steen, 1997 )
 Mathematics and Democracy: The Case for Quantitative Literacy (Steen, 2001 )
In an issues paper titled The Third R in Literacy prepared for the volume Quantitative Literacy: Why Numeracy Matters for Schools and Colleges, Randy Richardson and Dave McCallum (Richardson and McCallum, 2003 ) outline goals for quantitative literacy and provide examples of different approaches from a variety of institutions of higher education. Issues papers provide a variety of points of view on this topic. Discussion at the forum produced a whitepaper which is available for comment.
Bill Briggs,coauthor of Using and Understanding Mathematics, A Quantitative Reasoning Approach (Bennet and Briggs, 2002 ) provides an overview of quantitative literacy and other resources on his site Quantitative Literacy/Reasoning.
What is the role of the geosciences in contributing to quantitative literacy on campuses? What are the unique opportunities and challenges in the geosciences? How are geoscientists participating in campuswide initiatives in partnership with mathematics and other scientific disciplines?
What skills are important for all students?
The mathematics community has given considerable attention to the quantitative skills that are important for citizens. The Mathematical Association of America's Quantitative Reasoning for College Graduates: A Complement to the Standards (more info) recognizes the following goals which can serve as a starting point for work in the geosciences.
Every college graduate should be able to apply simple mathematical methods to the solution of realworld problems. A quantitatively literate college graduate should be able 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. Use arithmetical, algebraic, geometric and statistical methods to solve problems.

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

4. Recognize that mathematical and statistical methods have limits.
Two other publications provide important perspectives on the quantitative skills needed by citizens:
 Crossroads in Mathematics: Standards for introductory college mathematics before calculus. (more info) (American Mathematical Association of TwoYear Colleges)
 Principles and Standards in School Mathematics (more info) (developed by the National Council of Teachers of Mathematics for K12 education)
Interdisciplinary discussion of quantitative literacy is taking place on EXTEND: Perspectives on Mathematics Education (more info) , a national Internet forum on mathematics education. This forum is intended to involve new constituencies, to engage new voices, and to examine new perspectives in achieving quantitative literacy nationwide. Ongoing discussions surround access, expectations, articulation, integration, and numeracy.
What is the role of geoscience courses in developing these skillsat the introductory level? for future teachers? in the full suite of geoscience courses?
What skills are important for geoscience students?
While every faculty member and every employer will have a strong opinion about the skills needed for their research or employees, are certain skills important for all geoscience majors? For geoscience students pursuing graduate work? For those entering careers in environmental science, resource exploration or resource management?
While the role of quantitative skills in the geosciences is not a new topic (Shea, 1990 ), today's discussion takes place in the context of new advances in understanding how students learn and effective techniques for teaching mathematics. To capitalize on this work, and move forward in defining learning goals for geoscience students, identifying teaching methods that work, and sharing teaching materials, geoscientists need a common vocabulary and framework for discussing quantitative skills.
Preliminary discussion with several faculty suggests that an initial classification of quantitative skills taught in geosciences could include:
 Making measurements and representing data(graphs)
 Manipulating numbers (arithmetic, changing units)
 Using equations as language to describe, interpret and understand natural phenomena
 Making estimates
 Recognizing uncertainty and error (statistics)
 Using mathematical models and visualizations
 Recognizing assumptions and setting boundary conditions
An alternative approach is taken by Shah and West (Quantitative Skills Assessment In Geoscience Courses (more info) ), who have developed an assessment tool to determine what skills are being taught in geoscience courses. Widespread use of this tool could provide an inventory shedding light on our current practices.
What do we mean by "quantitative skills?" What are the roles of geoscience, mathematics, other science courses and research experiences in helping students gain these skills? How can our partnerships be strengthened to enhance student learning?
What can we learn from our sister disciplines?
As the importance and complexity of quantitative thinking becomes clearer, all scientific disciplines are considering their role in teaching quantitative literacy and the skills required for their majors. The geosciences can learn from their experiences.
The Mathematical Association of America in its Reports from the Curriculum Foundations Project (title altered) (more info) project has engaged science disciplines in discussions to help define its recommendations for the undergraduate mathematics curriculum.
Other disciplinary efforts include:
What aspects of these initiatives are transferable to geoscience? What can we learn from these efforts?
How will we know when we are effective?
First we need to know what we are trying to accomplish. However, as we proceed in setting our learning goals, the mathematics community has developed assessments for mathematical concepts that will be useful in evaluating learning. For example:
 Supporting Assessment in Undergraduate Mathematics (more info)
 Measuring What Counts: A Conceptual Guide for Mathematics Assessment (NRC, 1993 )
What are our goals? How can we measure our successes?