The Unique Role of Introductory Geology Courses in Teaching Quantitative Reasoning

Wayne Powell and David Leveson

wpowell@brooklyn.cuny.edu dleveson@brooklyn.cuny.edu

Submitted To: Journal of Geo-science Education, February 2002

ABSTRACT

Quantitative literacy (a basic familiarity with numbers, arithmetic and graphs) and quantitative reasoning (the application of logic to problems and the ability to understand the real-world meaning of numbers and mathematical statements) are end-members of a continuous spectrum of quantitative concepts. Different disciplines tend to lie on different points of this spectrum in terms of what they emphasize in introductory classes. Introductory geology lies mid-way on the quantitative spectrum. For example, probability is a concept that is essential to reasoning in geology, but commonly it is not expressed numerically. Rather, it is expressed and evaluated by appeal to logical argument. Weighing data and appreciating implications of scale, location and spatial relationships are critical in tackling geology problems, and one could argue that in a cross-curricular quantitative reasoning program these aspects of quantitative reasoning may be best delivered through introductory earth science courses. Thus, geological reasoning is a unique and valuable form of quantitative reasoning.

INTRODUCTION

As earth science teachers we hope not only to impart geological content knowledge to our students, but also to aid them in developing the critical reasoning skills needed to understand, interpret and apply knowledge in general. Facilitating the development of reasoning skills in geology is a difficult and complex undertaking, and includes acquiring general skills that are shared to varying degrees with other academic disciplines. Two basic, although not mutually exclusive, multidisciplinary elements of reasoning are: 1) literary reasoning, or the ability to understand and organize ideas through writing; and 2) quantitative reasoning, or the ability to understand and organize ideas through numbers and mathematical relationships.

In recognition of quantitative reasoning as a critical skill across academic disciplines, a faculty-training program (funded in part by a National Science Foundation Institutional Reform Grant) was established at Brooklyn College with the goal of introducing quantitative reasoning across the curriculum in low-level courses. With the help of released-time, about a dozen faculty members, including ourselves, met for two hours approximately twice a month over the Fall 2001 semester to discuss issues related to quantitative reasoning in a cross-curricular program. The group with which we were involved included faculty members from the departments of geology (our own department), biology, classics, computer and information science, economics, education, English, health and nutrition science, mathematics, physics, psychology, and speech.

This diverse cadre of educators brought a correspondingly diverse set of attitudes, beliefs and experiences related to the notion of quantitative reasoning. Unsurprisingly, there was little agreement as to what exactly ‘quantitative reasoning’ meant. People tended to subsume within the definition whatever approach they utilized or found appropriate for their particular area of expertise. In essence, each discipline rightly stressed the subset of quantitative reasoning skills that were in accordance with the demands of an introduction to their discipline.

The more numeric disciplines, such as physics and economics, stressed skills in algebraic manipulation, basic arithmetic, reading of graphs, evaluation of statistics and basic modeling. Additional skills, such as approximation, detection of logical fallacies, and verbal description of numerical relationships and their consequences tended to be stressed by the less numeric disciplines, including geology, biology, health science and speech. It is interesting to note that the American Association for the Advancement of Science’s “Science for All Americans” (Project 2061) states that while a person may not be expected to argue the technical aspects of science, everyone should be expected to “detect the symptoms of doubtful assertions and arguments”. Thus, less numeric disciplines have a critical role to play in the development of quantitative reasoning.

It became apparent within the first workshop meeting that different disciplines had distinctly different expectations and, therefore, different contributions to make to the overall development of quantitative reasoning in a cross-curricular approach. From our observations in the workshop at Brooklyn College, it was the less numeric disciplines, including geology, that tended to stress this reasoning skill in their introductory courses.

THE QUANTITATIVE SPECTRUM

The initial attempts of our workshop group to grapple with the definition of quantitative reasoning made it apparent that we commonly confuse two basic, distinct concepts: quantitative literacy and quantitative reasoning. Numerous researchers, including Cobb (1997) and Bennett and Briggs (1999) have defined these concepts. These two concepts are here defined as:

1) Quantitative Literacy: a basic familiarity with numbers, arithmetic and graphs. As in English literacy, it involves an understanding of the basic rules (grammar) of the language, in this case mathematics, and an ability to manipulate numbers.

2) Quantitative Reasoning: the application of logic to problems and the ability to understand the real world meaning of numbers and mathematical statements.

In our opinion, the concepts of quantitative literacy (QL) and quantitative reasoning (QR) are end-members of a continuous spectrum of quantitative concepts. To clarify the meaning and use of the QL-QR spectrum, let us apply it to an example that is probably familiar to most of us -- the graph of global average temperature from 1840 to 1990 (Figure 1a). This graph is a representation of complex real-world numeric data and as such could be used for many purposes in relation to a student’s development of quantitative skills. At the end of the spectrum towards QL (Figure 1b) lie basic skills such as being able to plot an X-Y line graph or being able to read data from the graph (e.g., what was the average global temperature in 1940 relative to today?). A notch toward QR might be the observation that the curve is irregular but is, on average, rising with time. A greater degree of reasoning would be required to extrapolate a continued increase in temperature in the future (based on the graph alone). Even further toward the QR end of the spectrum lie skills such as evaluation and questioning of data (Figure 1b). In this case for example, it might be asked, “how was the global average temperature calculated?” or “how reliable was the instrumentation?" Finally, the ultimate example of QR is the ability to apply quantitative/numeric data in the development of a logical argument.

Figure 1. Examples of the spectrum between quantitative literacy skills and quantitative reasoning skills with respect to a specific graph. (A) Example of a global average temperature graph from 1840 to 1990. (B) Quantitative skills referring to the global temperature graph, positioned on the QL-QR spectrum.

We noted during the term’s set of workshops that there was a tendency for the approaches and emphases of different disciplines to occupy different parts of the QL-QR spectrum. Certain disciplines find it useful to deal with an idealized or reducible world, one that can be meaningfully described with quantitative statements. (In mechanics, it may be useful to ignore friction or assume the possibility of an absolute vacuum.) Faculty that belong to such disciplines tend to place a greater emphasis on QL in the teaching of introductory or non-majors courses. For them, mathematics is the underlying language of the discipline; therefore, QL must be a prelude to QR. An analogous scenario would present itself in an introductory English class: to express ideas usefully in prose, the prose must be lucid. A student’s paper may enfold brilliant thoughts, but the paper is likely to receive a poor grade if they are obscured by egregious grammatical errors. Only after the student achieves some mastery of the English language can the emphasis be shifted to critical analysis of ideas. Similarly, in physics, critical analysis of ideas can come only after the student achieves mastery of the mathematical language.

Members of disciplines for which idealization and quantification are unwarranted, impractical or impossible either tend to emphasize QR over QL in their introductory and non-majors courses, or not include a quantitative component at all. A brief summary of where we perceived various disciplines to lie on the spectrum with respect to their emphasis on teaching in introductory courses, based on the responses and statements of workshop participants, is summarized in figure 2.

Figure 2. Perceived location of various disciplines on the QL-QR spectrum in reference to their teaching emphasis in introductory courses, if quantitative concepts are discussed.

GEOLOGY AND QUANTITATIVE REASONING

Introductory geology falls within the center of the QL-QR spectrum. Whereas there are numerous sub-disciplines within the earth sciences that are highly numeric (geophysics, geochronology, hydrology) typical introductory courses in physical and historical geology deal with a world that is complex and chaotic, irreducible to algorithms. Variables are uncountable. The rock record is characterized as much by what is absent as by what is physically there. Processes may have operated in the past of which we have little or no knowledge. And yet despite these enormous limitations, geologists do propose hypotheses as to patterns and relationships, even if they are not commonly quantifiable. How may such hypotheses be judged? In theory, hypotheses are compared and evaluated on the basis of their perceived probability. Is geology successful? If success is judged in terms of fulfilled predictions, geology is remarkably successful.

Probability in geology commonly is not expressed numerically. Rather, it is expressed and evaluated by appeal to logical argument. Accordingly, in the typical introductory geology classroom an instructor can justifiably emphasize approximation and reasoning. Gaining the ability to approximate reasonably may demonstrate more understanding than the mechanical attainment of exact answers. In geology, rather than memorizing or manipulating equations, students should learn to weigh evidence (often circumstantial) and judge likelihoods. That is not to say that an introductory geology student must be completely shielded from the world of algebra, but mathematical tools should only be employed to demonstrate better the character of geological reasoning, or on occasion, its elegance.

Additional aspects of quantitative reasoning that present themselves commonly in the earth sciences are spatial analysis and geometric analysis, particularly in terms of three-dimensional geometric relationships. Estimation of the modal abundance of minerals in a rock, interpretation of contours on a map, characterization of folds and faults, basic relative dating techniques such as the application of the Law of Superposition and the Law of Cross-Cutting Relationships are all applications of the spatial and/or geometric reasoning that is at the heart of geology.

QUANTITATIVE REASONING IN GEOLOGY – AN EXAMPLE

As geologists, we have become so familiar with geological modes of argument that we generally employ them instinctively when we encounter a new problem and reason through it. We can easily lose sight of the many assumptions and logical steps that we make in even the simplest of geological problems. Essentially, we use our experience (geological bias) to weigh likely scenarios instantly and arrive at conclusions quickly. It is, however, these basic steps and assumptions that are at the heart of geological reasoning and so these elementary steps are the ones that would be most valuable to introduce to our introductory students as unique forms of quantitative reasoning.

Our argument as to the nature and role of quantitative reasoning in geology is best illustrated by means of a real field exercise employed in our non-majors introductory course. As part of the Brooklyn College required science sequence, all students must take a one-semester introductory geology course. Part of the course requirement is a field trip to Central Park, where there are numerous large outcrops of complexly deformed schists and gneisses cut by aplite and pegmatite dikes. On one particular horizontal, low relief rock surface, the student sees two intersecting dikes: a crude ‘X’ pattern in which an aplite is present as two slightly offset segments separated by a pegmatite (Figure 3). The students, as geologists, want to solve a historical problem: what was the sequence of events that created the ‘X’ pattern. Students typically propose one of two hypotheses: (a) the ‘Aplite First’ hypothesis, in which older aplite is cut by younger pegmatite; (b) the ‘Pegmatite First’ hypothesis, in which the pegmatite acts as a barrier to younger aplite intrusions.

Figure 3. Intersecting aplite (A) and pegmatite (P) dikes. The two aplite segments are offset across the pegmatite and if extended, do not line up.

The ‘Law of Cross-Cutting Relationships’ furnishes a historical narrative by applying a geometric criterion: if one structure cuts another, the one that is cut formed before the one that did the cutting. Upon cursory examination of the ‘X’, students commonly perceive that the pegmatite passes uninterrupted through the aplite, dividing the latter into two segments (Figure 4). Advocates of the ‘Aplite First’ hypothesis propose the following sequence of events:

Figure 4. Intersecting aplite (A) and pegmatite (P) dikes as they would appear without offset of the aplite segments across the pegmatite.

First, the aplite was formed as a single, continuous body; later, the pegmatite formed, cutting across the aplite. To test this hypothesis, they predict that the two aplite segments will be seen to “line up.” If the prediction is fulfilled, the “Aplite First” hypothesis is supported. However, when the students examine the outcrop more carefully, they see that the prediction is not fulfilled: the two segments do not “line up” (Figure 3). But the hypothesis need not be abandoned. If they take into account three-dimensional geometry, an appropriate modification may be proposed: that the pegmatite intrusion was accompanied by ‘dilation’ perpendicular to the direction of intrusion (Figure 5a and b). (The instructor may have to point out the more subtle requirement that the two dikes be vertical) (Figure 5c). The students predict that lines connecting ‘congruent’ points on the aplite segments will be found to be perpendicular to the length of the pegmatite (Figure 5d and e). In the case of the Central Park outcrop, the students’ observations confirm this prediction and the modified “Aplite First” hypothesis is supported.

From the very beginning, while part of the group busies themselves with the development of the argument articulated above, another group focuses on the apparent mismatch of the two aplite segments. In consequence, they pursue another line of reasoning. They propose a “Pegmatite First” hypothesis in which the pegmatite formed first; then, at a later time, an aplite intrusion advanced toward the pegmatite but was unable to progress through or beyond it (Figure 6a). The pegmatite dike served as a barrier to the advancing aplite. Then, at a still later time, another aplite intrusion advanced from the other side toward the pegmatite (Figure 6b). The pegmatite again served as a barrier and stopped the second aplite intrusion from progressing any further. That is, they reason, the pegmatite-aplite contact is not “cross-cutting” and the Law of Cross-Cutting Relationships does not apply.

Figure 5. Modified ‘Aplite-First’ hypothesis. (a) Aplite prior to pegmatite intrusion. Future path of pegmatite intrusion shown by dashed line. (b) Dilation due to pegmatite intrusion. (c) Three-dimensional view of Figure 5b showing vertical dip of dikes. (d) Connection of congruent points of the aplite-pegmatite junction. (e) Three-dimensional view of Figure 5d.

Figure 6. 'Pegmatite-First' hypothesis. (a) Intrusion of first aplite dike. (b) Intrusion of second aplite dike.

When the two groups exchange ideas, the Aplite Firsters point out that if the pegmatite came first, then it would be a strange coincidence if lines joining ‘congruent’ points of the aplite-pegmatite junction were perpendicular to the length of the pegmatite. On the other hand, if the aplite came first, the geometric relationships of the lines would, of necessity be as described, if the dikes are vertical and were separated by dilation. Thus, in making that argument, the Aplite Firsters are resorting to an analysis of probability.

At this point, the students are confronted with two competing hypotheses, one of which seems more probable than the other, but neither of which can be entirely ruled out. The instructor points out that such a dilemma is commonly faced by geologists. They have to make decisions despite uncertainty. When asked upon what basis they would decide, the students opt for relative probability.

The instructor now asks the students how their ‘most probable’ choice might be tested. The students suggest employing laboratory procedures to determine the absolute ages of the pegmatite and the two aplite segments. If the “Aplite First” hypothesis is correct, the aplite segments must be the same age, and they must be older than the pegmatite. In this practical exercise in field geology, students eventually see the potential value of numerical analysis, but only as a final step in a long, essentially non-numeric problem.

DISCUSSION

In the example exercise above, the students engaged in reasoning that primarily involved the logical analysis of geometric relationships and probability. Such reasoning is certainly quantitative although mostly non-numeric. Geological reasoning commonly involves visual and even tactile experiences. The involvement of the two senses makes the thought process less abstract. Hopefully, such incorporation of concrete, sensory data will aid in development of QR skills. This should be particularly true for visual and tactile learners who may suffer in problem solving situations that involve abstract and numeric concepts.

Quantitative reasoning in introductory geology classes generally focuses on determining the probability of alternative explanations. Hypotheses that gain acceptance are those that are either likely in and of themselves (e.g., the intersecting dike problem described above), or those which benefit from the likelihood of more extensive claims within which they are subsumed (e.g., the ‘X’ pattern developing as a result of the evolution of a magma that allowed intrusion of different liquids over time) or to which they are linked (e.g., intrusive activity related to orogenic events). For a world which, of necessity, must be described in terms of multiple variables, incomplete data, and imprecise measurement, assessment of ideas based on their probability is both reasonable and pragmatic. This is more in line with Pollock’s (1997) observation regarding the reality of quantitative reasoning in real-world situations: “Instead of ‘Here’s a problem, solve it,’ in real life it’s ‘Here’s a situation, think about it.’”

Geology is certainly not the only discipline that must confront and interpret a vast number of inter-related variables. Ecological, epidemiological and sociological problems certainly also must deal with confounders (additional variables that affect the variable being studied, thereby potentially leading to incorrect or incomplete interpretation). What sets geology apart from these other disciplines, in terms of quantitative reasoning, is its historical nature. In theory, it would be possible in field studies of on-going relationships, such as ecological interconnections in Central Park, to continue to collect more and more data until the confounders are resolved. However, in the ‘X’ pattern example discussed earlier it would be impossible to resolve the issue with absolute certainly because it happened in the past and without being observed. Geologists accept the idea of problems without absolute answers. It is inherent to the nature of the problems we study.

Real world problems rarely have a “correct answer.” Furthermore, rarely is a complete set of data available to the problem solver due to limitations in time or resources. Accordingly, becoming skilled in assessment based on estimation and comparison of probability would be especially valuable to students as they face problems in real-world situations. This reasoning skill would enhance their understanding of the natural world and, equally importantly, it would provide them with tools that they might use to navigate the political, economic and social universes of their everyday lives. This would be especially true if their quantitative reasoning skills had been rounded out by the acquisition of a broad set of specialized skills (e.g., algebraic manipulation) emphasized in the other disciplines encountered by students in a liberal arts program.

REFERENCES

American Association for the Advancement of Science Project 2061, 1989, Science for All Americans.

Bennett, J.O., and Briggs, W.L., 1999, Using and Understanding Mathematics -

A Quantitative Reasoning Approach, Addison-Wesley, New York.

Cobb, G.W., 1997, Mere literacy is not enough. In: Why Numbers Count - Quantitative

Literacy for Tomorrow's America, edited by Lynn Arthur Steen, p. 75-90.

New York : College Entrance Examination Board.

Pollock, H.O., 1997, Solving problems in the real world. In: Why Numbers Count -

Quantitative Literacy for Tomorrow's America, edited by Lynn Arthur Steen,

p. 91-105. New York: College Entrance Examination Board.