Building the Quantitative Skills of Students in Geoscience Courses

Introductory geology courses taught from a question-based approach that effectively incorporates problem solving demonstrate to students that science is more than a collection of facts. By having students put together observations and calculations to answer questions about the Earth, the course provides opportunities for students to develop more quantitative ways of thinking. Proficiency with quantitative problem solving comes from doing in-class exercises, homework, and exams that include numerical and graphical problems requiring arithmetic, algebra, and geometry. Incorporating quantitative problem solving is hampered by student perceptions about geology courses as well as the lack of introductory geology textbooks with a quantitative focus. However, quantitative materials can be successfully incorporated into large introductory geology courses if the instructor is accessible, engaging, and positive towards students' problem-solving ability.

Diversity in mathematical preparation is common in quantitative geoscience courses, such as geophysics and hydrology. One way to handle this diversity is to design a sequence of homework assignments in which the mathematical difficulty increases progressively ("stepped homework"). The sequence of assignments for a typical quantitative course should include the following steps: 1) "plug-ins," 2) algebraic manipulation, 3) graphing, 4) trigonometry and logarithms, 5) multistep problems, and 6) calculus and computer spreadsheets. Examples of problems from an introductory geophysics course are provided for each step. To be effective, this sequence must be coupled with ample opportunity for students who have difficulty to obtain assistance. Possible sources of assistance include tutoring by the instructor, working in a recitation section, and tutoring by peers. In a geophysics course that begins with seismology, a stepped homework sequence can be essentially completed by the end of that unit, leaving the students better prepared for success in the remainder of the course.

Problems that involve quantitative skills require students to reason logically. Strong logicalreasoning and problem-solving skills are necessary for students in proceeding with the application of mathematical methods. introductory geoscience courses offer an excellent opportunity for students, especially nonscience majors, to develop their problem-solving and quantitative skills. A series of logic problems that increase in difficulty level as the course progresses are given to students in my introductory geoscience courses. Because each student works through problems at his/her pace, many exercises are given as homework. To tie the problems to the course material, real geologic data are used. For example, one logic problem uses streamflow discharge data for a local river. Each student must perform unit conversions, solve the problem, plot data, and provide a written interpretation of the graph. At the end of one semester, a majority of students state that their self-reported ability to solve logic problems has increased. It is essential that students have the confidence and ability to solve relatively simple problems before they can go on to complete problems requiring the use of more complex quantitative methods.

We have modified our introductory geoscience course by adding self-contained problem sets dealing with isostatic rebound, flood recurrence intervals, and geochronology. Students had access to faculty and undergraduate-student TAs, but no class or lab time was devoted to the problems or their solutions. The students produced formal reports, with their responses supported by tables and graphs. We found that the students were generally undaunted by the problem sets, even when they introduced advanced mathematical concepts and required substantial data manipulation. Subsequent exam questions suggested they developed an understanding of the concepts covered by the problem sets.

We have used a variety of approaches to introduce quantitative concepts into our introductory geology classes, some of which rely on using case studies of local rivers to improve student interest and appreciation of quantitative methods in scientific problem solving. Fortunately, there are rivers in most parts of the country, so these approaches are easily transferable and applicable in many areas. A simplified form of the continuity equation applied to stream flow serves as the starting point for developing a discussion of how and why local rivers are channelized for flood control. Students also analyze the recurrence interval of flooding along a local river of interest.

Density is one of the most fundamental and integrative topics within the physical sciences and the Earth/space sciences. However, the concept of density cannot be assumed to be understood by typical non-science majors at the collegiate level. Three activities that investigate the densities of common solids, liquids, and air were designed to help these students more fully understand this fundamental and critical concept. Students investigate and determine the densities of three equal masses of aluminum foil that have been molded into three different volumes. They determine the masses and volumes of different amounts of water, calcite, and isopropyl alcohol. They graph this data to find the slope of three lines and determine the densities of each substance. Students partially evacuate a flask and determine the density of air using the change in mass of the flask and the volume of water required to refill the flask. The three related investigations have proven to be excellent experiences that enable these students to become competent in their knowledge about density and more confident in their ability to apply this knowledge to more abstract Earth/space science processes. The investigations have also provided excellent opportunities to foster quantitative reasoning in nonscience majors.

Students build quantitative and analytical skills as they determine rock density in the laboratory. They measure dimensions, calculate volume, measure mass, and calculate density for eight rock specimens. In the next laboratory meeting, they analyze errors, correct data, and report rock-density values based upon the collective efforts of the class. These data are used again, later in the semester. Compressional velocities of seismic waves are calculated in lecture using the densities for granite and metabasalt. Students calculate the porosity of three sandstones in a later laboratory exercise. Students determine differences in elevation between the continental masses and oceanic basins attributable to densities of continental and oceanic crust in another laboratory exercise. These exercises are effective at building understanding of concepts and quantitative skills, based upon the interpretation of essay-exam scores, laboratory-instructor observations, and student comments.

Global Positioning System (GPS) receivers are a type of scientific equipment that is accessible to the general public and potentially capable of motivating students to learn. Inexpensive receivers provide students with experience aimed at developing spatial awareness and practicing mathematical skills as they acquire data and analyze it through calculation and mapping. Here I report on an experimental use of GPS in a small honors class, with the goal of eventually applying it more broadly in introductory courses. GPS was incorporated into an existing physical-geology course in the form of six laboratory and homework exercises (for example, marking positions with latitude and longitude, closing a triangle, drawing maps) with positive results. Students enjoyed using the technology, appreciated working outside the classroom, and learned the basics of GPS, while developing directional and spatial awareness and quantitative skills in trigonometry, graphing, and use of computer spreadsheets. Successful introduction of GPS into this small class augurs well for its use in larger generaleducation courses.

Although the ability to use a geologic compass is an important skill for geologists and can be a useful and stimulating skill for any student, most introductory geology courses do not include any work on the use of the compass. This is unfortunate since compass work, introduced early, can help improve the quantitative skills of all students by giving them hands-on experience at solving practical problems. The equipment needed is readily available at relatively low cost and a variety of compass-based exercises can be designed to strengthen student skills. Worthwhile exercises include the following topics: quadrant and azimuth notation, magnetic declination, compasstraverse closure, elevation measurement, distance between two landmarks, and strike and dip.

A mock trial in which undergraduates serve as expert witnesses and law students serve as their attorneys is an effective vehicle for developing quantitative skills and enhancing written and oral communication skills. I have developed an interdisciplinary course based on the book A Civil Action. The book deals with the legal struggle of families in Woburn, Massachusetts, who sued two corporations alleging that improperly handled industrial chemicals entered the groundwater system, were captured by two municipal wells, and prolonged ingestion of the contaminated water caused leukemias and other health disorders. Students analyze aerial photographs, well logs, streamflow records, permeability tests, and water-level and water-quality data from the trial to complete assignments that become exhibits in the mock trial. Assignments include construction of geologic cross sections, potentiometric maps, hydrographs, flood recurrence graphs, and calculation of hydraulic gradients, groundwater velocities, and contaminant travel times. Trial transcripts and newspaper articles serve as background materials for a term paper. Based on the computational assignments, background readings, and a discussion of professional ethics, students compose an expert opinion from the viewpoint of their client and are deposed by opposing counsel. A jury of undergraduates is impaneled for the one-day mock trial in which the law students make opening statements and closing arguments, and conduct direct examinations and cross examinations of the scientific experts. The course teaches students how to develop and defend their opinions, how to question the opinions of others, the limitations of data collection and analysis, and the importance of integrating computational and communication skills.

Scientists commonly gather data and develop equations to describe relationships among data and variables using linear regression. Providing geoscience majors opportunities to determine physical relationships using regression techniques is important for their understanding of the nature of science. Fortunately, regressions are easily calculated with spreadsheet or statistics software, and, if the mathematical basis is well developed, students can understand the predictive power of a regression and apply it to many problems.

In the activity presented here, undergraduate geoscience majors use linear regression techniques to determine rates of Pacific-plate motion over the Hawaiian hotspot through time. Using age and location data for the Hawaiian-Emperor volcanic chain, students calculate the rate of plate motion for the entire chain and the separate components, then determine whether plate motion has been constant over time. Using latitude and longitude data, they determine the location of the bend in the volcanic chain. Finally, they develop a relationship between age and location to make predictions about where existing volcanoes will lie in the future and the age of the bend in the volcanic chains. Students are introduced to error analysis by examining data errors and learning about the sources of those errors and by evaluating formal errors calculated in the regression analysis.

Forging a link between quantitative skills and geologic problem solving is a valuable instructional approach that can guide the development of quantitative course material. A conceptual framework is presented that shows how various research tasks (for example, data collection and hypothesis development) employ certain combinations of quantitative skills (for example, graphical presentation and algebra). The framework shows how this "repertoire" of skills can be explored and strengthened by posing course assignments as research problems. An example problem, "What is the major source of nitrogen to the South Fork of the Palouse River?," illustrates implementation of all the tasks and skills in the framework via a four-week unit of coursework. Smaller units can focus on subsets of tasks and skills. Experience with units on structural geology, igneous petrology, hydrogeology, and isotope geochemistry suggests that the framework can be applied to virtually any geoscience topic at any level in the undergraduate curriculum.

A data-intensive research project on the climate history of Africa has been made part of an introductory college-level earth-science course. The project introduces students to the challenge of working with real data as student teams analyze and interpret extended records of monthly rainfall and average temperature for stations across Africa with the goal of defining regional climate patterns and assessing evidence regarding climate change. The exercise emphasizes quantitative skills in data manipulation and statistical analysis within a framework of critical thinking, decision making, oral and written communication, and collaborative participation. The social context of the project is apparent immediately to the students, but the scientific context requires additional input from the instructor. The project has been carried out in classes with up to 100 non-science students and is a successful vehicle for engaging African-American students in the geosciences. It can also readily be adapted for use in high-school earth-science classes.

Students in a surficial-processes class helped design and completed a research project to measure the weathering rates of marble tombstones at two locations in southwestern Montana. Measuring the rates of rapid geomorphic processes emphasizes the quantitative aspects of research project design, data collection, and analysis. The project reinforced students' recently acquired knowledge of statistics (from a required math class) as they used it, at their own initiative, to solve a geologic problem. When the field data were more complex than they expected, students demonstrated for themselves that mathematical analysis could give meaning to their data.

We offer a course that our majors enroll in concurrently with their calculus course that "shadows" the topics covered by the calculus professor. In the shadow course, the students work collaboratively to build mathematical skills and apply calculus concepts to solve geoscience problems provided by the instructor (two examples are given). Students earn a grade of "pass" by demonstrating their involvement in learning. It is not feasible to assess statistically the effect of the shadow course on students' grades in the calculus courses; however, the shadow course has led to other positive outcomes. Our majors have developed more positive attitudes toward mathematics, including calculus, and get the message that geoscience faculty value and actively support their learning of calculus. We have gained insight into our students' strengths and weaknesses in mathematics that may help us to incorporate mathematics more effectively into geoscience courses. The shadow course is helping to foster active collaboration among geoscience and mathematics faculty that strengthens the teaching of scientific problem-solving using quantitative methods in both the calculus and the shadow courses.

Computational Geology is a spreadsheetintensive, geological-mathematical problemsolving course recently developed at the University of South Florida. Requested by nontraditional students and now a required part of the geology curriculum, the course finishes off the required calculus sequence and its prerequisites. It makes connections between the various strands of mathematics and between mathematics and geology. It aims to enhance mathematical literacy and computational skills and to improve the mathematical comfort level of our students. It also promotes a mathematical problem-solving disposition that is useful to students regardless of whether they remain in geology.

One approach to teaching quantitative skills to all students in a department is to construct a matrix of the desired quantitative skills versus courses in the departmental curriculum. Faculty members complete the matrix by listing the assignments in each course that build each particular skill and then design activities or assignments that either fill a gap or more fully develop a skill or application. Faculty members discuss the matrix on a regular basis to report progress and challenges, share ideas with each other, and plan future directions. This iterative process enhances the quantitative skills of students by incorporating quantitative activities and problems throughout the geoscience curriculum. When some quantitative work is included in every departmental course, students recognize that quantitative tools are important in the geosciences. Communication, cooperation, and planning at the department level and regular reviews of the matrix are key aspects for developing quantitative skills across the departmental curriculum.

Analysis of stratigraphic sections typically consists of recognition and interpretation of lateral and vertical heterogeneities in sedimentary rock. Qualitatively, significant information is obtained by careful observation of changes in various lithologic components (grain composition, size, texture, sorting) as well as the presence or absence of a wide range of sedimentary structures (ripples, crossstratification, desiccation cracks). Taken together, these physical manifestations of conditions within the depositional environment allow for construction of complexly detailed facies models of ancient sedimentary systems. However, modern stratigraphic analysis is becoming increasingly concerned with more than the construction of facies models. Such transcendent analytical effort represents a further refinement of past attempts at quantification of processes of deposition. To date, the principal approaches to quantitative stratigraphic analysis have been statistical—and the datatype upon which the most effort has been placed is bed thickness. As such, evaluation of several commonly used analytical techniques provides an important background to, and overview of, the statistical analysis of bed thicknesses. By providing students with an introduction to the statistical foundations of modern stratigraphy, it is possible to greatly enhance understanding of both stratal architecture as well as the relationships between depositional process and the stratigraphic record.

The motion of open ocean waves and associated water-particle orbits constitute a focused topic that occupies a lasting niche in our arsenal of simple, pedagogically compelling things to offer introductory students in the geosciences. The topic provides a vehicle for introducing the general subject of wave behavior, a universally important phenomenon in the geosciences; it challenges students to understand counterintuitive ideas regarding the difference between wave and water-particle motions, serves as a base for advanced topics (for example, oscillatory ripple formation), and is readily "accessible" to students in that wave motions and particle orbits can be easily demonstrated or "tested" in either lab or field conditions. But unless students are exposed to this topic in a manner that goes well beyond introductory-text explanations, possibly in advanced courses, they are not apt to gain an understanding of ocean-wave behavior beyond that provided by purely kinematic explanations. Because of the pedagogical importance of gravity waves—of which ocean waves are an example—and because ambiguities exist in current introductory-text descriptions of waves, we assemble and summarize a dynamical explanation of their behavior. A notable bonus of this explanation is that it provides a simple, concise introduction to the ideas of buoyancy and gravitational forces—also universally important subjects in the geosciences—and how these forces interact during wave motion.

Specifically, the orbit of a water particle beneath a train of ocean waves involves a clockwise motion when viewed from a perspective where the waves are moving from left to right. The vertical component of this motion is associated dynamically with pressure fluctuations about the average pressure state such that the buoyancy force exceeds the gravitational force beneath wave troughs, and the gravitational force exceeds the buoyancy force beneath wave crests. The horizontal component of the motion is associated with these pressure fluctuations wherein the pressure decreases horizontally in the direction of wave motion between wave crests and their leading troughs, and the pressure increases horizontally in the direction of wave motion between wave troughs and their leading crests. These forces alternately accelerate and decelerate the water with simultaneous vertical and horizontal components leading to approximately circular orbits. Peak water (orbit) speeds decrease with increasing wave speed. The attenuation of orbits with increasing depth is related to downward attenuation of the fluctuating pressure field, rather than being attributable to frictional damping.

This dynamical explanation can be concisely presented at several levels. The first involves a qualitative description that appeals to graphical representations of the pressure and velocity fields beneath waves. The second appeals to a simple trigonometric representation of waves and the simplest possible form of Euler's equations combined with the approximation of hydrostatic conditions. The third involves solving linearized forms of Euler's equations.

Teaching the concepts and fundamentals of hydrology and hydrogeology, particularly to undergraduates who have limited field experience, presents a challenge. Many innovations from working models to computer simulations have been used to improve visualization of the subject and to add a practical applied component to training. Recently, as a result of a partnership with private firms, two monitoring wells were completed on the University of Wisconsin-Green Bay campus to augment several previously installed porous-cup lysimeters. These enabled a graduate student to sample vadose and ground and surface water from a variety of streams, ponds, and seeps and to characterize the surface and subsurface water resources of the 700-acre campus. The investigation revealed a number of relationships such as the high sulfate concentrations in one of the monitoring wells, elevated concentrations of chloride in several surface water bodies, and an unexpectedly low concentration of ions in a pond on the campus golf course.

The results of the study, and the convenient sampling sites available on the campus, are being used to strengthen the science curriculum. Students now have opportunities to obtain practical experience with water-quality sampling and analysis instruments and to improve their applied technical skills. They may also develop an appreciation of the difficulties involved with field sampling and measurement, and an understanding of the importance of proper handling of samples and accountability issues of sample collection and analysis.

Understanding how water is transported and stored in the subsurface is a difficult concept for introductory earth-science students. We have developed a hydrology minicourse that integrates field and laboratory experiences to help undergraduate students gain a better understanding of ground-water flow in aquifers. The centerpiece of the minicourse is an investigative field trip that permits analysis of a local aquifer that provides drinking water for the university community. Students collect qualitative and quantitative field data on grain size, thickness, and geometry of different stratigraphic horizons within the aquifer and then construct a small-scale laboratory model of the aquifer using boundary conditions determined from the field investigation. The aquifer model allows students to test hypotheses of ground-water flow by conducting a series of modeling experiments. The experiments test questions such as: "What is the influence of porosity and permeability on groundwater flow?" and "What is the effect of regional dip on ground-water flow?" Analysis of pre- and post-minicourse examinations demonstrates that students are able to better communicate fundamental hydrologic concepts after completing the minicourse.