ONLINE EXTRA: Quantitative Reasoning in the Geoscience Classroom: Modeling Functions and Logarithmic Scales
VICTOR J. RICCHEZZA (firstname.lastname@example.org) is a doctoral candidate and H.L. VACHER is a Professor of Geology at the University of South Florida School of Geosciences
Have you ever tried presenting graphs to your students only to experience frustration when they look first to the data points, ignoring important information on the graphical axes? Does this frustration lead to a less quantitative presentation of your course—do you leave the graphs (or the math behind them) out entirely? Geoscience courses are often viewed as being qualitative, despite the fact that modern geoscientists practice in a thoroughly quantitative field (Manduca et al., 2008). Enhancing our students' skills and experience in quantitative reasoning in undergraduate geology courses can be difficult, but it is essential if students are to work successfully in the profession after graduation (Manduca et al., 2008; Vacher, 2012). Geoscience courses offer a precious opportunity to present mathematics in context (Wenner et al., 2009), which we cannot let pass us by. Two topics that fit very nicely together, and are quite relevant to geosciences, are logarithmic scales and graphing modeling functions. In this article we discuss a unit from our course, and how it teaches these concepts.
In addition to mathematics courses, geology students at the University of South Florida (USF) are required to take a course to improve application of quantitative skills to geoscience. One popular option is a course in Computational Geology (GLY 4866, hereafter CG). The CG course covers a variety of mathematics topics in geologic context, with integrated problem sets, worked examples, and lab activities (Ricchezza, 2016; Ricchezza and Vacher, 2016, 2017; Vacher, 2000b).
The CG course uses problem sets to improve quantitative skills. The problem sets include homework packets with worked examples of the week's new topic(s) and multiple problems to be worked through by the start of the next week's class. One sample problem, given the first week of class, says "My office is on the fifth floor of SCA. There are two flights of stairs between floors, and 12 steps per flight. When you walk up the stairs to my office from the parking lot, how many stairs do you climb?" (Answer at end of article.) Students complete ten of these weekly sets, then take a quiz on the material (to encourage completion of the problem sets). In the second of the two class sessions each week, students meet with the graduate teaching assistant (TA) for a lab, during which they perform practical operations related to the week's problem set (Fig. 1).
Among the CG topics of interest to a wide (interdisciplinary) geoscience audience is logarithmic scales, including their relationship to modeling functions. We have found that a fundamental weakness our students have in the general area of quantitative literacy (QL) is the ability to "see" and grasp deeply information communicated in a graph. It is through teaching logarithmic scales and modeling functions that we hope to address this common deficiency (Vacher, 1998a, b, 1999, 2000a, 2003, 2004, 2005). (See http://nagt.org/nagt/jge/columns/compgeo.html for additional columns related to this course.)
Problem Set 8 (of 10) is Logarithms and Log Scales (see Fig. 2), an essential topic for seismology and geochemistry (among a variety of other areas of practice in and out of the geosciences). The lab for this problem set/topic is "Slide Rules and Log Scales." For those born a bit too recently to have experienced their use the first time around, logarithmic slide rules were used for centuries as a calculation aid and were essential to such science and engineering tasks as putting humans on the Moon and returning them safely to Earth. Slide rules make use of the properties of logarithmic scales and the rules for their interactions to assist the user in performing calculations (for a brief history of slide rules, a tutorial on their usage, and a virtual slide rule application, see http://sliderulemuseum.com/SR_Course.htm ). To complete this lab, students are provided with logarithmic slide rules that were produced on 3D printers (the 3D-printed slide ruler template can be accessed free at http://www.thingiverse.com/thing:2305300) at the USF Alliance for Integrated Spatial Technologies (AIST), with labels from a paper-based slide rule printed on blank labels rather than standard paper and applied to the plastic sliders (our slide rules were labeled with a paper slide rule model developed by Scientific American https://www.scientificamerican.com/media/pdf/Slide_rule.pdf). Students complete 2-4 calculation problems using the slide rule for each type of calculation available (multiplication, division, squaring, square root, cubing, cube root, reciprocal, sine, tangent, and common logarithm), with some calculation problems chosen to require additional thought. That is, logs on the slide rule can be done only for base 10 (common logs), so students were given a log in a different base; likewise, logs on the slide rule can be done only for numbers up to 10, so students were given values above 10 to require use of rules of manipulating logarithms in order to complete the problem. Additional issues arise in the placement of decimal points and the location of zeroes, as log scales on a slide rule generally run from "1" to "1," and we require the students to understand why. Students learn through this experience that the slide rule was (and still is!) a useful tool, but that it did not and does not think for them. Figure 3 shows the 3D-printed slide rule.
The final portion of the Slide Rules and Log Scales lab is completed by students being handed two sheets of standard graph paper and being tasked with constructing two log scales and using these scales to construct a standard multiplication/division slide rule. Successful students are able to complete the final portion of the assignment properly if they understand how log scales work and how they relate to arithmetic scales (such as those on standard graph paper).
Returning to the learning goal, a novice learner will often find the eyes drawn to the data points (or curves) on a plot. In contrast, a more experienced learner, such as, we hope, a veteran of this course, will look first at the axes of the graph. Figures 4a and 4b show an example of why the 'eyes-to-data' view is problematic and the 'eyes-to-axes' view is helpful. The seismic energy, plotted versus the earthquake magnitude, produces two very different-looking plots, depending whether one uses standard arithmetic graph axes or semilog paper (that is, one arithmetic axis and one logarithmic). Additionally, the "standard" plot (see Fig. 4a) shows little to no visible difference between most of the data points until magnitudes 8 and 9 are reached – in fact all of the energy values for magnitudes 7 and below appear to be zero, which is not true – correct determination of the energy readings from this graph is nearly impossible. Finding the energy reading on the semilog plot is simple... provided the learner knows how to read a logarithmic scale (see Fig 4b). [Note: this particular pair of graphical plots deals with energy released through seismic waves from earthquakes. For those unfamiliar, earthquakes release energy when sudden movement occurs inside the Earth, and this energy travels as seismic waves. For more information on earthquakes, see United States Geologic Survey resources at https://earthquake.usgs.gov/learn/ and https://pubs.usgs.gov/gip/earthq1/.]
This learning experience is followed two weeks later by the final problem set and lab (see Fig. 5), focused on four standard modeling functions (linear, logarithmic, exponential, and power functions). As with the prior problem sets, students complete the set on their own, take a quiz, have the instructor work through the quiz and fundamentals of the set, and review worked versions of the set and quiz. Students then meet for the final lab where they graph modeling functions. In this lab, students are given two ordered pairs of (x,y) coordinates. They are then given two x-values, with one falling between the two given points, and one outside. Students are provided with four sheets of different graph paper: standard (arithmetic) graph paper, semi-log paper (with log scale on the x-axis), semi-log paper (with log scale on the y-axis), and log-log paper. They are then asked to plot the two ordered pairs on each different set of graph paper, draw an apparent straight line through those points, work out the equation of the "line" (i.e., the function), and find the two missing y-values. Based on their experience during the problem set and the prior class session, the students should know that these four sets of graph paper, if they appear to show a straight line, actually represent linear, logarithmic, exponential, and power functions, respectively, and, moreover, that they have been provided with the necessary information to find all that has been requested of them in the lab.
What's the point of all this? As noted earlier, when students enter our course, they make the mistake many students (and indeed many university graduates) make when reading a graph: the first thing they look at, aside from perhaps a title, is the cloud of points or trendline in the data. By the conclusion of the CG course, our hope is that students first look is at the axes of the graph, so that by the time they look at the actual data they have already preset their minds to analyze the sort of functions and data sets that might be framed by variations in the axis scales and labels. A USF geoscience alumnus who took this course wouldn't see just a series of points arranged in a straight line – they'd (for example) see a logarithmic scale on the y-axis and an arithmetic scale on the x-axis and rightly judge that "straight line" to mean an exponential function (see Figure 4b). While this specific goal won't be the same for all geoscience courses, and certainly not for all educational levels, the goal of quantitative literacy is something reasonable and attainable for the courses on which we all work, and we hope this set of activities from our course gives you some ideas you can use to make your courses more quantitative.
Answers to sample problems:
Problem 1: Note that in the building (SCA), the ground floor is actually floor 1, so the correct answer is 4 intervals X 2flights/interval X 12steps/interval = 96. Many students will incorrectly place the number of intervals at 5, failing to account for the starting number of 1.
Figure 2: a. 12 yard line. b. 90 yard line
Figure 5: a. this is an exponential function where E rises 3 log cycles for every 2 levels of M. b. Joules: log𝐸𝐽=1.5 𝑀+4.8.; tons TNT: log𝐸tons=1.5 𝑀−4.8. c. value of E at magnitude 0 is 6.3 x 104 joules, and increases to 31.6 times as much for each step. Value of E at magnitude 0 is 1.6 x 10-5 tons of TNT and increases by the same multiple. d. Magnitude 7.2
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