Hint: Not what schoolchildren studying math and science in school do all day.
It is a common supposition that mathematicians and scientists mostly do a lot of calculating and computing, always getting the Right Answer (except for the conspiracy theories about “Darwinism” and Global Warming). Calculating in search of Right Answers is certainly the main part of the arithmetic, algebra, and so on up to calculus that we teach in schools, in every subject except plane geometry, which depends on proof. It is also most of the physics curriculum, though less so in chemistry and much less so in biology.
There is substantial calculation and computing within these professions, and in engineering, too, but it is a tool, not the defining part of their work. What scientists and mathematicians do most essentially is to notice patterns, also known as symmetries, and the relationships among patterns, and then set out to find more patterns and more relationships, and to prove, where possible, that these relationships are unavoidable and necessary in the nature of things. Then they try everything they can think of to tear down what they hope they have proved, and only accept what survives this process.
Engineers, on the other hand, mostly make tradeoffs, attempting to optimize their design decisions based on what management or the customer wants, and what current scientific knowledge and technology permit. It is an old engineering maxim:
Good, fast, and cheap: Pick two.
Thus a good and fast computer will not be cheap; a good and cheap car will not be fast; and a fast and cheap printer will not be any good.
Building bridges is the accepted archetype for engineering in general, going back in particular to the Roman bridges and aqueducts that still stand more than 2,000 years later. In most of the 19th century, bridges not made of stone commonly broke and fell down, because nobody knew how to predict the forces they would experience. John A. Roebling succeeded in building bridges that stayed up, in spite of weather, sometimes inferior materials, and corruption in the governments paying for the bridge building, by calculating how strong their bridges had to be within the knowledge and capabilities of the time, and then making them six times stronger.
The Brooklyn Bridge, the largest built in the 19th century, was designed by John Roebling and built under the supervision of his son, Washington Roebling. It is both a famous and an infamous example in the annals of engineering, for the beauty of its design, because it has stood for more than a century, and because of the corruption of Tammany Hall under Boss Tweed during its construction. When it was discovered that inferior iron [sic; steel was still too expensive] wire had been supplied for the bridge cables, it turned out that the bad wire could not be removed, and so it was necessary to recalculate how big the cables needed to be, and force the fraudulent wire company to supply all of the extra wire needed at no further charge. It was also necessary to change the method of inspecting incoming shipments from the manufacturer to prevent further fraud.
Oh, that’s another thing engineers have to understand, to the extent possible. How to work the political and legal systems, or corporate and government management systems, in order to be able to do their jobs properly. What did not happen in the BP oil rig blowout and fire and so on, where BP had ordered the drilling company to replace the drilling mud in the bore, which was keeping oil and gas from coming to the surface, with much lighter seawater, among many other safety violations. Or the Hubble Space Telescope contretemps, where Perkin-Elmer misconfigured the testing equipment for the telescope mirror it had just ground, and NASA failed to supervise the project sufficiently. Or again NASA management overruling engineers to launch the Challenger Space Shuttle when the temperature was below its safe operating range, resulting in its complete destruction and the deaths of the entire crew. Or the refusal to fix the levees in New Orleans. Or…never mind. There are many books about such disasters, including catastrophes that ended civilizations. We have other tofu to fry. ^_^
The US National Academy of Sciences is nearing completion of A Framework for K-12 Science Education: Practices, Crosscutting Concepts, and Core Ideas, with an emphasis on what scientists and engineers really do. They have published a draft version on the Web.
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We consider eight practices to be essential elements of the K-12 science and engineering curriculum:
- Asking questions (for science) and defining problems (for engineering)
- Developing and using models
- Planning and carrying out investigations
- Analyzing and interpreting data
- Using mathematics, information and computer technology, and computational thinking
- Constructing explanations (for science) and designing solutions (for engineering)
- Engaging in argument from evidence
- Obtaining, evaluating, and communicating information
Calculation and computing come in mainly in practices 2, 4, and 5. I would add several other practices, particularly collaboration, looking for patterns, thinking of ways to “falsify” their conjectures and theories, and keeping detailed notes on the progress of their investigations, including every idea for falsifying the conjecture or theory under investigation. And, as noted above, fighting with management.
Collaboration should go without saying. Whether it is an experimenter following up on the work of a theoretician, or the hundreds or thousands of people in a great astronomical observatory or collider facility, or just professors mentoring graduate students, collaboration is fundamental. We must also remember
If I have seen farther than others, it is because I have stood on the shoulders of giants.—Isaac Newton
Here is a pattern made of smaller patterns, one in each column.
This overall pattern relates to several larger mathematical patterns, such as Figurate Numbers, Conic Sections (and thus Projective Geometry), Finite Differences, Calculus, and more, and to Galileo’s discoveries about gravity and the nature of physical laws.
Here is a very different pattern. Archimedes noticed one day that the amount of water overflowing a bathtub was directly related to how much of his body he had put in, thus solving the problem of how to measure volumes without all sorts of fiddly calculations. Legend has it that he ran home from the bathhouse naked, shouting “I found it!” because that was what he needed to solve a different problem set by the King of Syracuse.
I could easily multiply examples, such as Alexander Fleming noticing a circle of dead bacteria in one of his cultures, and asking, “Why?” rather than throwing it out. However, I prefer to turn to another of the NAS best practices, that is, asking good questions. The best questions inherently do not have right answers. Of course, many really bad questions don’t have right answers, either. The problem with questions that do have Right Answers is that overemphasizing them teaches students to be helpless, in several directions.
- When children don’t have the Right Answer, and don’t know how to get it, but the teacher has it in the Teachers’ Manual that accompanies a textbook, the children get the idea that only teachers and other authority figures can get right answers, primarily because they already have them. This was intentional in the Prussian education system and those of other empires modeled on that, of course.
- When the teacher gives the Right Answer, but it makes no sense to the child, the child may have nowhere to turn, and so just gives up. Among my favorite incorrect Right Answers are “variable numbers”, “You can’t add apples and oranges,” and the supposed “long”and “short” vowels in English. Latin and Japanese have actual long vowels, as in “Lūgēte, Ō Venerēs Cupīdinēsque,” (Mourn, O Venuses and Cupids) or “konpyuutaa” (computer) and “ookii” (big). English has diphthongs, as in “aisle”, “they”, “throw”, and letters and combinations used for more than one sound as in “book”, “boot”; “back”, “bake”; “hot”, “hotel”, sometimes in combination with the “silent e” rule.
- The most difficult are the official Right Answers imposed by state religions and ideologies of one kind or another, which have kept humanity in thrall for millennia of recorded history, and are still trying. North Korea is the nastiest of the survivors of this custom, but by no means the only such disaster area. In some cases it is clear that the ruling classes are motivated simply by greed and lust for power, and do not actually believe what they force others to proclaim loudly and often.
The notion of free inquiry, not bound by mythology, seems to have arisen among the Ionian Greeks, particularly Thales of Miletus. They can be said to have invented the foundational ideas of science and formal mathematics (beyond arithmetic and special cases in geometry) out of earlier cultural materials and informal discoveries of Egyptians and Babylonians. Unfortunately, their works were apparently suppressed by the mystical, aristocratic Pythagoreans and Platonists, so that we have only fragments of their teachings, including an early version of atomic theory (Democritus), a heliocentric astronomy (Aristarchos), and other doctrines now taken for granted (though in very different forms). They also created a variety of preliminary notions of what constitutes scientific method, and thus of what scientists, and therefore schoolchildren, should do all day.