"I Don't Have A Clue"- A Format for Introducing Problem Solving in the Earth Sciences

DAVID S. CHAPMAN is a Distinguished Professor Emeritus for the Department of Geology and Geophysics and Dean Emeritus for the Graduate School at The University of Utah- Salt Lake City.

Sitting in an apartment in Uzes, France, while on sabbatical, I was reading an International Herald Tribune story about a California couple who tried for a month to feed themselves on a dollar a day per person. The slant of the article was towards the difficulty of affording healthful food, but it contained a figure on the average amount of money that a North American spends, per day, on food.

I turned to my wife Inga and said, "What do you think is the average cost of feeding oneself for a day?" "I don't have a clue," she replied, rather too quickly. That response pushed a button for me. One that is often pushed by students who give an identical response when queried about something they could figure out.

So I responded, "No clue, eh? Does it cost 10 cents a day?" "It costs much more than that," said Inga. "Does it cost $100 a day?," I continued. "Don't be ridiculous," she said. I went on to explain that I was not trying to be ridiculous, but was in fact trying to make the point that she did have a clue about the daily cost of food. She had in fact quickly bounded her estimate by two numbers.  After some further thought about our grocery bills she said, "If you need a single number, then 8 dollars a day."  The number given in the International Herald Tribune story was 7 dollars a day.

I think this anecdote illustrates that the phrase "I don't have a clue" is used too often to avoid having to work out the details.  In the classroom setting, it perhaps means "Ask someone else" or "Go away."  But all individuals have stored knowledge about many things.  They are just not accustomed, or often asked, to pull together pieces of information from a variety of sources in order to solve a problem they had not remembered solving in school, or perhaps for which there is not a single answer.

Estimating answers to problems by relating the known to the unknown and bounding answers with quantitative arguments is a fundamental aspect of science.

This article describes a setting and a series of questions I have used at a weekend retreat with university students to stimulate problem solving in the Earth Sciences.

The setting and objectives

For a number of years I was engaged with an NSF program, GK-12, which put science graduate students in the K-12 classroom.  The GK-12 Fellows acted as science resource persons for teachers, assisted with field trips and science fairs, and served as inspirational models for students who might consider science as a career.  Our program at the University of Utah was a collaborative program with the Utah Museum of Natural History and the Salt Lake School District.  Our program was called WEST (an acronym for Water, the Environment, Science, and Teaching), and drew graduate students from three university departments: Geology and Geophysics, Atmospheric Sciences, and the ecology and evolution side of the Department of Biology.  It was important for all of our GK-12 fellows to develop good communication skills and good teamwork skills.

At our annual September retreat at a Wasatch Mountain ski resort, not yet open to the general public, we spent two and a half days working with teachers on classroom management skills, learning about inquiry-based teaching, and developing field exercises that would be suitable for various grade levels.  The final morning, before the graduate students were treated to a Sunday brunch at another of the resorts, they experienced a problem solving session that they claimed was totally new for them.

The students were divided into groups of three or four, and each group was provided a flip chart and magic markers. Their calculators and smartphones were removed, and they were given a series of problems - one at a time.  They were generally given 15 or 20 minutes to solve each problem as a group.  A moderator moved among the tables, asking questions to push groups who were hung up, but otherwise just collecting observations.  Occasionally, when groups were stumped because no one knew a required number, for example the price of water, the moderator assisted with information.  At the end of the time, each group made a presentation to the assembled groups, giving its answer to the problem, indicating the logic tree in solving the problem, and justifying the many assumptions that have to be made.

After hearing all of the groups, the moderator compiled and compared the various answers, probed about logic paths that were followed, and asked what information would be necessary to create tighter bounds on the answers and if, or how, that information might be gathered.  The initial question often keyed follow-on discussions.  The mood was light, and positive.

Problem 1. Rubber tire                           
Problem 2a.  Dilution                             
Problem 2b.  Sea level rise   
Problem 3. Conservation and costs              
Problem 4. Value of water                      
Problem 5. Gas stations in Iowa              
Problem 6. How much oil?                      
Problem 7. How long will the oil last?              
Discussion                        

Problem 1. Rubber tire

A very useful first problem is borrowed from the book "Consider a Spherical Cow" (Harte, 1985).   Problem number one each year has been, "What thickness of rubber is worn off car tires with one revolution of the tire?"

This is a good first question because most, if not all, would be tempted to say "I don't have a clue."  It is not a question that we are likely to have thought about.  Nor is it a question that we intuitively have an answer for, except to say "not very much."  But it is a problem that most people have knowledge for and, with a few minutes of thinking and back of the envelope scribbling, can come up with an answer.

Most car owners or drivers know, for example, that an average set of tires lasts for about 50,000 miles, at least from claims by tire advertisements. We also know roughly the thickness of treads when the tires are new, say 5-10 mm, and that the tire should be replaced before the "tread is bare."   The final information needed is the diameter of a typical wheel, which automotive specialists know in detail, but no one has trouble setting their hand at a level above the floor to show how big the average wheel is.  The number of tire revolutions needed to roll the tire 50,000 miles is easily calculated from the circumference of the tire and, soon after, the thickness of rubber worn off by a single tire revolution.

At this point, nearly every student "gets it."  They realize that they have reached into their archive of general knowledge, used a geometric relationship such as the circumference of a circle, and performed simple division without a calculator or computer to come up with a reasonable answer.  In doing so within a group, they have had to decide whether a tire gets, 40,000, 50,000 or 70,000 miles, or whether the difference matters, as well as what a typical statement such as "my tires come up to my knees" translates into in terms of quantitative measure.  They also learn, after some false starts, the convenience of using scientific notation and working in powers of ten.

I am always impressed at how much learning about problem solving can emerge from such an innocuous problem.  That is why it has remained problem number one.

Problem 2a.  Dilution

A second problem is about big numbers, not a small number like the thickness of rubber worn off in one revolution of an automobile tire.  In fact the big number has a name: Avogadro's Number. The problem can be introduced in the context of an old and abandoned maxim that "the solution to pollution is dilution," or in terms of the questionable use of oceans as a dumping ground for pollution.

The question is:  "If I tagged all the water molecules in a 1 liter bottle, dumped that water into the oceans, waited until the oceans were thoroughly mixed, and refilled the bottle, how many of the original molecules would I capture?"  This problem has a fairly simple logic tree, but requires several estimates, and definitely promotes the use of powers of ten in the calculation.

The dilution problem requires one to know how many water molecules are in a liter and the volume of the oceans.  I have been impressed by how many students actually remember that Avogadro's number is about 6 x 1023 molecules in a mole, often to three decimal places.  Science graduate students generally have little problem with figuring out that there are a little more than 50 moles of water molecules in a liter.

The volume of water in the oceans is more difficult.  It seems well known that about 70% of the Earth is covered by ocean but more challenging to convert that knowledge into a quantitative area in square meters.  The surface area of Earth is required and there are several ways that students have come up with that estimate: geophysics students have had drilled into them that the Earth's radius is 6371 km, others might know that the distance from the equator to the poles (one quarter of the Earth's circumference) is 10,000 km, or that one degree of latitude is 112 km.  More than once, a group has argued that the distance from the equator to the pole is not very different than a transatlantic or transpacific flight taking about 10 hours, and that planes fly at 500 miles per hour or 800 km per hour.  Those values produce an estimate of 8000 km for the equator-pole distance, good enough for the dilution problem.  Student groups that are stumped can be led to a good estimate through a series of hints.

Geology students generally know that the average depth of water in the oceans is between 2.7 km, the depth of water at an oceanic ridge, and 5 or 6 km, the depth of water in abyssal plains.  The average depth of water in the oceans is 3.8 km. Some probing from the moderator, however, and a few hints, usually extract reasonable bounding limits of one and ten kilometers.

Once the volume of oceans is known, the dilution factor and thus the number of the original molecules recovered in the resampled liter is easily calculated.  Students are surprised by how large a number the answer is, about 20,000 of the original molecules.  But that surprise probably comes from not being able to comprehend the magnitude of Avogadro's Number.

Follow up discussion of this problem can be rich, questioning the wisdom that dilution could ever be the solution to pollution, the likelihood and mechanisms of mixing in the oceans, an estimate of mixing times, ocean currents and their effect on mixing and climate, the effect of continental drift to change the configuration of oceans and either promote or prohibit mixing, and more.  Because there are more problems to solve in these sessions, the topics are rarely discussed in detail, sometimes just listed as related topics for further discussion.

Problem 2b.  Sea level rise

A very good follow-on question to the ocean dilution problem, with much more societal relevance, is "How much would sea level rise if the ice sheets on Greenland and Antarctica were to melt completely?"  Students could use the surface area of the oceans already calculated for the dilution problem, so they would only have to estimate the volume of ice to be melted and then spread that melt water uniformly over the world's oceans. They might also address whether or not the small difference in ice and water densities was sufficient to ignore.

If a globe were to be available, students could map Greenland and Antarctica onto a sheet of paper and estimate what fraction of the area of the globe is covered by those two land masses – a few percent.   An alternative approach would be to make use of the widely known fact that about 30% of the area of the planet is made up of continents.  If the seven continents were of equal size, then the average area of a continet would be one seventh of 30%, or about 4% of the surface area of the planet.  Antarctica and Greenland together would be the size of a smaller continent, although bigger than Australia, so 3% might be a good estimate.  The actual number is 3.1%.

The thickness of ice could be bracketed as more than a kilometer, possibly by those who followed Greenland and Antarctica drilling reports, but not more than several kilometers. Actual ice thicknesses are 1.5 km for Greenland and 2.2 km for Antarctica.

This sea level problem also lends itself to another important strategy in problem solving – working out an algebraic solution prior to plugging in numbers or performing a calculation.  Equating a volume of ice (area x thickness) that would be spread over the world's oceans (area x thickness) leads to a simple equation.  If the area of Antarctica plus Greenland and the area of the oceans are both expressed as fractions of the Earth's surface area, one further step leads to the approximation that sea level rise is about 1/25 of the average ice thickness.  Bracketing ice thickness between one and two kilometers leads to a predicted sea level rise of 40 and 80 m.  Best estimates of Antarctic ice sheet melting would increase sea level by 60 m, Greenland by 6 m, and although not considered here, mountain glaciers by 4 m.

"The estimates of sea level rise are an entry to discussing what fraction of the world's population lives below 70 m; what agricultural land would be lost?; what major cities would be abandoned?" etc.

The process of making a numerical estimate of sea level rise seems to make much more of an impression than simply being told a number.

Problem 3. Conservation and costs

One year I borrowed a card from my resort room that stated: "Using towels more than once saves hundreds of pounds of detergent and thousands of gallons of water each year." And went on, as many hotels do today, to request that one hangs up towels that can be reused and have sheets changed only after the complete stay.  I simply turned the ecological statement into a series of quantitative questions. How many pounds of detergent in a year?  How many thousands of gallons of water a year?  Make an estimate for this resort and for this ski town.  Estimate not only the detergent and water saved, but the money saved as well.

Our students were enthusiastic about this problem, reflecting the environmental inclination of many students in the earth and atmospheric sciences.  I was pleasantly surprised by how quickly groups were able to estimate the volume of water and liquid detergent required to launder towels and sheets for one room in a day.  Attaching costs to the washing was more problematic.  Occasionally groups would estimate the cost if one were to take the laundry loads to a laundromat, with which they could relate.  We had to supply the cost of water but stayed out of a debate among groups whether or not they should include the energy cost of heating the water.

Scaling up to the town of Alta was relatively simple.  They could count the number of rooms in the Peruvian Lodge where we were staying, and estimate the number of rooms in the other three lodges in the resort.  Healthy debates about occupancy rates for various times of the year arose, but the differences were not appreciable, and students appreciated that for this exercise, emphasis should be on general principles.  Details would be important if one were tasked as an environmental consultant to save the Peruvian Lodge money, much less to save the world.  The rough estimates indicated considerable savings for the resorts in Alta.

Problem 4.  Value of water

Another year, the WEST Fellows and school teachers were enlightened by University of Utah Professor William T. Parry in an informative session on the mining history of the town of Alta, UT.  They learned, for example, that silver was discovered in the area in 1864.  Alta was established as a town in 1871, with a population of 3,000, 180 buildings, and 26 saloons.  Between 1864 and 1930, $150 million were removed from this mining region in the form of silver, gold, and zinc.  The history of Alta is rich in the joys of mineral discovery, fraud, and despair when the ore ran out.  By 1930, Alta had become a ghost town, with only six registered voters.  A second life for Alta was started in 1938 with the construction of the first ski lift; Alta is now one of the premier ski areas of the world.

But water is also a valuable resource, not only for Alta, but in the arid West in general.  Seeing potential for motivation from the historical session, and to emphasize the importance of water, I created the following problem.  "Estimate the annual value of water that falls in the Little Cottonwood drainage basin (eg. Alta) and runs in Little Cottonwood Creek down to the Salt Lake City water treatment plant.  Compare the value of this water to the much heralded value of ore extracted from the Alta mines."

Several estimates were required, including the area of the drainage basin and the annual precipitation.  The Little Cottonwood Canyon drainage basin is well defined, but the question required students to visualize the geometry of the watershed and put numbers on the dimensions of the drainage basin in which they were gathered.  Alta receives most of its precipitation in the form of winter snow and skiers in the class knew that Alta receives about 500 inches of powder snow each winter.  There was a meaningful discussion amongst the students, especially between the meteorologists and geologists about what percentage of the snowfall ablates, what the water equivalent of 500 inches of powder is, how much runs off as surface water in the Little Cottonwood Creek, and how much percolates into the ground and forms deep groundwater.  Students were reassured when told that many of these water partitions are imprecisely known, and that they should simply make an informed estimate and be able to justify the assumptions that went into their estimate.  If students had use of smart phones or computers they would simply look up the stream flow in Little Cottonwood Creek, but that would have avoided the estimating processes that were involved in the "I don't have a clue" exercise.

No one, including me, knew offhand the pricing of water in the Salt Lake district, so that value was looked up and shared.  Salt Lake City Corporation charges about $1 per 100 cubic ft of water, or 40 cents per cubic meter.  The annual value of water captured in Little Cottonwood Canyon surprised all of us: $16M, slightly greater than the peak ore production in the 1870s not accounting for inflation. The implications of the value of fresh water however escaped no one, especially as a factor in the continuing tussle between environmentalists and developers in the canyons of the Wasatch Mountains bounding Salt Lake City.

"Finding that the current price of water was $1 per 100 cubic ft or about 0.04 cents per liter, students often asked "why do we pay $2.00 for a liter of bottled water?"

"Now that's a good question," I always answered.

Problem 5.  Gas stations in Iowa

The last three questions involve gas and oil, a subject of immense importance to our society.  The first is a fun, open-ended question about gas stations that encourages creative approaches and multiple options to making quantitative estimates.  The second reaches into geologic and geographic knowledge bases, and is linked to the concept of a finite resource.  The third prepares students to think about societal transition to a time when petroleum is limited.

A wildcard question I have repeatedly used for reasons that will emerge, is "How many gas stations are there in Iowa?"  The question was suggested by a geochemist colleague who grew up in Iowa City.  Other states would work, though perhaps not as well.

Unlike the rubber wear on a tire, which has a definite logic tree for the solution, solving the gas stations in Iowa problem has several approaches, and student groups over the years have discovered many of them.  Discussion and debate within teams is therefore rich.  Again, students realize that they have to move from the known to the unknown.

A popular approach to answering this question is the "geography approach."  Iowa, the argument goes, is a farming state, made up of small towns. We know that towns of 5,000 to 10,000 people generally have more than one, but not many, gas stations.  How many towns of that size exist in Iowa? This approach requires knowing the population of Iowa.  When asked for this number, one group argued that the average state population is 300 million divided into 50 states.  Iowa was not one of the populous states such as California or New York.  Nor was it a low population state such as Wyoming.  Just a medium to small state with a medium to small population of about 3 million.

A variant on the 'number of small towns geography' approach was the freeway gambit, offered by a graduate student who hailed from Iowa.  She knew that two interstate highways bisect the state, I-35 which runs north-south and I-80 which runs east-west.  Freeway exits occur about every 20 miles, she argued, and each freeway exit has at least two gas stations.  The small town algorithm should be used to fill in the spaces not served by the freeways, she argued.  This approach requires the north-south and east-west distances across the state, less familiar information to most than the driving time to cross the state, which of course can be easily related to distance with an appropriate assumption.  There are also larger cities that must be dealt with.

An original approach, developed one year by a creative group thinking outside the box, used an economic tactic.  Starting with an estimate of the population of Iowa, this group then estimated the number of drivers and/or the number of cars in the state.  Knowing the average miles driven per year by the average US driver, and the average fuel efficiency of an average American car, this group calculated the amount of gas used by Iowa residents in a year.  With an assumption of the profit in selling a gallon of gas, they estimated the total annual gas sales revenue in the state.  And with another economic assumption, they arrived at the number of gas stations that the gas sales economy could support.  Perhaps surprisingly, for the year that the economic approach was advanced, it lead to an answer within a factor of two of the geographic approach answers.

Whichever approach is followed, the gas stations in Iowa problem has always lead to similar, reasonable answers with considerable latitude for imagination. Why is Iowa a good state to use?  Except in Iowa and neighboring states, there are unlikely to be many in the class who hail from Iowa. Students who are asked about gas stations in their own state tend to try the census approach because they know the number of gas stations on various highways.  Problem solving is more than a census count.  Iowa is large enough, but not too large.  It is a state that many people have driven through.

While "gas stations in Iowa" may seem to be a wildcard problem in otherwise Earth Science based problems, it is the topic most mentioned when students who have participated in the past are asked about the retreat.

Problem 6.  How much oil?

It is not surprising that a question as general as, "How much oil is left in the world?" would generate the most discussion, and sometimes disagreements, among any of the previous questions.  It is a question that experts disagree vehemently about and panels of scientists spend good parts of their careers working out the details.  But it is also a question that, at a basic level, is amenable to a back of the envelope treatment.

The geologic knowledge required for this question is information about sedimentary basins, or the percentage of continents and continental margins covered by basins that could host oil bearing strata.  Not all sediments contain hydrocarbons to generate oil so some regions can be discounted.  Discussions also focus on both the generation of oil and the storage of oil.  Both require assumptions about conditions at depth – temperature to control oil generation and porosity to govern storage.

In twenty minutes, groups are generally just warming to this problem, but it is one that continues on after the formal sessions are over.

"And even if the answers student groups determine are far from the current expert answer on how much oil we have, a very important lesson is learned. Oil resources are finite."

Problem 7.  How long will the oil last?

A logical follow-on question to "How much oil?" is "How long will the oil last?"  This question addresses a more pressing environmental issue of the consumption and uses of petroleum.  A good strategy is for students to focus on the US oil consumption, and then to make a rough estimate of how many US equivalents there are in the world.  Now the problem is more tractable.  It can be based on the US population, the rough number of drivers, average miles driven per year, and average consumption rates for cars and trucks.  Then one must estimate the fraction of petroleum that is consumed by cars and trucks, for air travel, and for heating and cooling and other industrial processes.  Just as the lesson learned in estimating "how much oil?" is that the resource is finite, the lesson learned in estimating "how long will the oil last?" is a length of time measured in timespans of their own lives- and it critically depends on our consumption rate.

Discussion

Articles and books on problem solving are neither new nor necessarily novel.  Among relatively recent books, "Consider a Spherical Cow" (Harte, 1985) is perhaps the best known. The term "spherical cow," in fact, is much quoted when describing various approaches to estimating solutions to problems, particularly in the environmental sciences.  A follow-on book (Harte, 2001) introduces us to the "cylindrical cow." Two more recent books (Weinstein and Adam, 2008; Mahajan, 2014) describe systematic approaches to problem solving; they grew out of university courses in physics departments.  These books include sections dealing with numbers- both large and small, unit conversions, and useful techniques for dividing difficult problems into simpler parts (divide and conquer)- what Mahajan (2014) calls a toolbox of techniques.

The art of making quantitative estimates is also developed masterfully throughout MacKay's (2009) book "Sustainable Energy – Without the Hot Air."  Virtually every page contains estimates of quantities or back of the envelope calculations as he asks, and then systematically answers, the question "Can a country (the UK) provide 100% of its energy requirements from sustainable resources – wind, solar, biomass, hydro, wave, tide, and geothermal?" The mantra in MacKay's book is: "We need numbers not adjectives."

In contrast to these books, this article does not attempt to cover all techniques of problem solving, nor does the scope qualify for an entire course, or even part of a course.  But for those who want to introduce quantitative estimating and problem solving, this paper identifies a retreat or seminar setting and a format that takes little time and has many benefits.  Several aspects of these retreat sessions are important: (1) working in groups, preferably with members coming from different disciplines or sub-disciplines; (2) presenting group results and arguments to the entire body; (3) comparing the different group results- not only the estimates, but also the logic trees followed; and (4) discussing follow-on or related questions.

The specific problems addressed in this article demonstrate how the problems can be chosen to fit local geographic situations, disciplinary issues, items of current interest, or simply intellectual challenges.  The problems I have used and described in this article grade from introductory questions, to difficult but tractable questions, and to serious societal questions (sea level rise, value of water, how much oil).  There is sometimes a need to step back and say, "This is an exercise on learning how to solve problems by relating knowns to unknown, not in solving the weighty problems of the world."  Although talent to do the former would help with the latter.   Nor should this type of problem solving seminar be restricted to science students or graduate students.   I once devoted one session in a Liberal Education class, "Global Environmental Issues," to such problem solving with an interesting consequence.  A subset of about eight students, not any of them geology majors, were so taken with the ideas that we met regularly after class once a week for coffee and discussion for the rest of the semester.  I was instructed to bring one problem to each coffee session, which would be discussed and solved in half an hour.

Overall, students involved in these problem solving sessions have reported considerable satisfaction with the sessions and have indicated that the sessions have changed their own way of looking at problems.  They also report that the last thing they say when asked a question is "I don't have a clue."

References

Harte, J., 1985, Consider a Spherical Cow.  A Course in Environmental Problem Solving: Los Altos, CA, William Kaufmann, Inc.

Harte, J., 2001, Consider a Cylindrical Cow: More Adventures in Environmental Problem Solving. Sausalito: University Science Books.

MacKay, D.J.C., 2009, Sustainable Energy – Without the Hot Air. Cambridge: UIT Cambridge Ltd.

Mahajan, S., 2014, The Art of Insight in Science and Engineering: Mastering Complexity.  Cambridge: MIT Press.

Weinstein, L., and Adam, J.A., 2008, Guesstimation. Princeton: Princeton University Press.




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