In Extreme Ownership: How U.S. Navy SEALs Lead and Win, Jocko Willink and Leif Babin explain the importance of getting team members to believe in the mission. As a math teacher, this is one of my greatest challenges. In the words of Dan Meyer, Chief Academic Officer at Desmos, and a prominent pundit on math education, in his breakout TED Talk from 2010, as a math teacher
“I sell a product to a market that doesn’t want it but is forced by law to buy it.”
In other words, the team doesn’t believe in the mission. Dan (we have met) talked about his ways of getting them to believe in the mission. My favorite way, so far, is to engage students in the story of math and the people who discovered or invented it (that dichotomy is itself a fun or frustrating debate, depending on your perspective).
My reading program now consists of three different books: The Professional Chef (9th edition), the textbook for The Culinary Institute of America; Physics with Applications (6th edition), by Giancoli; and Math Through the Ages: A Gentle History for Teachers and Others (Expanded Edition), by William Berlinghoff and Fernando Gouvêa, published in 2004 by Oxton House Publishers and the Mathematical Association of America.
For this post, I’m quoting (rather extensively) the latter, as its ideas may help my fellow math teachers looking for hooks for lessons or even the year’s courses. I won’t be doing the one-and-done book report I’ve done for other books because I expect to take a slower pace through this book, and the other tomes I mentioned above are going to slow me down (and that’s okay with me, obviously).
Here are my highlights from the chapter titled “History of Mathematics in a Large Nutshell,” particularly the first two sections: “Beginnings” and “Greek Mathematics.”
- Mathematics developed with writing:
“No one quite knows when and how mathematics began. What we do know is that in every civilization that developed writing we also find evidence for some level of mathematical knowledge.”
- We detect the development of mathematics by applications:
“It became important to know the size of fields, the volume of baskets, the number of workers needed for a particular task. Units of measure, which had sprung up in a haphazard way, created many conversion problems that sometimes involved difficult arithmetic. Inheritance laws also created interesting mathematical problems. Dealing with all of these issues was the specialty of the ‘scribes.’ These were usually professional civil servants who could write and solve simple mathematical problems. Mathematics as a subject was born in the scribal traditions and the scribal schools.”
- We’ve found different evidence from different cultures, and then, as now, the hike from Egypt to Iraq was apparently not a popular trip for scribes:
“… we have only a few documents that hint at what Egyptian mathematics was like….The situation with respect to the cultures of Mesopotamia is quite different….These two civilizations existed at about the same time, but there seems to be little evidence that either influenced the other’s mathematics.”
- Egyptian math (and learning it) was not terribly different from the way we see math today (which I think has more to say about us than the Egyptians):
“The Egyptian mathematics of 4,000 years ago was already a fairly well-developed body of knowledge with content very similar to some of what we learn about calculation and geometry in elementary and high school today. It was recorded and taught by means of problems that were intended as examples to be imitated.”
- The Babylonians (remember, we know more about their math than about the math of Egypt) built math around the problems of government and management, and, after the work was done, the Babylonians went back for MORE:
“The mathematical activity of the Babylonian scribes seems to have arisen from the everyday necessities of running a central government. Then, int he context of the scribal schools, people became interested in the subject for its own sake, pushing the problems and techniques beyond what was strictly practical. Like a musician who is not satisfied with playing at weddings and graduations, the well-trained mathematical scribe wanted to go beyond everyday problems to something more elaborate and sophisticated. The goal was to be a mathematical virtuoso, able to handle impressive and complex problems.”
- Despite the current preeminence of ethnic Chinese in mathematics, we have little to go on regarding the ancient development of the subject at the other end of the Silk Road:
“We do not know a lot about very early Chinese mathematics….The mathematical texts we do have seem to reflect the rise of a class of civil servants who were expected to be able to solve simple mathematical problems. Like the texts from Egypt and Mesopotamia, they contain problems and solutions. In China, however, the solutions are often presented together with a general recipe for solving this type of problem.”
- What of the other Cradle of Civilization, you ask? Well you might:
“We know even less about early Indian mathematics. There is evidence of a workable number system used for astronomical and other calculations and of a practical interest in elementary geometry.”
- The mathematics of China did not affect the Western development of the subject much, but:
“The Indian mathematical tradition influenced Western mathematics quite directly.”
- Now we get to the meat of the early history of math, the culture that built the Mediterranean network of cultures in rivalry with the Phoenicians and, thanks to Roman victory over Carthage, wrote the history of math and many other things which dominates the story today:
“Many ancient cultures developed various kinds of mathematics, but the Greek mathematicians were unique in putting logical reasoning and proof at the center of the subject. By doing so, they changed forever what it means to do mathematics. We do not know exactly when the Greeks began to think about mathematics. Their own histories say that the earliest mathematical arguments go back to 600 B.C.. The Greek mathematical tradition remained a living and growing endeavor until about 400 A.D..”
- It is the cultural hegemony created by Greek colonization and the military might of its successor civilization, Rome, that makes Greek mathematics what we consider it to be:
“It is important to stress that when one speaks of “Greek mathematics” the main reference of the word “Greek” is the language in which it is written. Greek was one of the common languages of much of the Mediterranean world. It was the language of commerce and culture, spoken by all educated people. Similarly, the Greek mathematical tradition was the dominant form of theoretical mathematics.”
The rest of my reading at this point (the above and below quotes are from pages 6-24) makes a nice list which I have used for several years to build a historical reference framework of names and stories in my classes. The Greeks that make the cut are:
- Thales (circa 600 B.C.) —
“… the first person to attempt to prove some geometrical theorems, including the statements that the sum of the angles in any triangle is equal to two right angles, the sides of similar triangles are proportional, and a circle is bisected by any of its diameters.”
- Pythagoras (circa 500 B.C.) —
“Most scholars believe that Pythagoras himself was not an active mathematician … [Pythagoreans] seem to have been much concerned with the properties of whole numbers and the study of ratios (which they related to music). In geometry, they are, of course, credited with the Pythagorean Theorem. … It is likely, however, that the most important success often credited to the Pythagoreans is the discover of incommensurable ratios (my note: this led directly to the idea of irrational numbers).”
- Euclid (circa 300 B.C.) —
“What we have are his writings, of which the most famous is a book called Elements. It is a collection of the most important mathematical results of the Greek tradition, organized in a systematic fashion and presented as a formal deductive science. The presentation is dry and efficient.”
- Archimedes (circa 250 B.C.) —
“Archimedes wrote about areas and volumes of various curved figures …”
- Apollonius (circa 200 B.C.) —
“…Apollonius wrote a treatise on conic sections that is still an impressive display of geometric prowess.”
The book then delves into a favorite topic of mine: the three great geometric problems of antiquity: the quadrature of the circle, the trisection of the angle, and the duplication of the cube. This is a favorite theme of mine when students are primed to discuss how and why people do mathematics, meaning discovering, proving, teaching, and publishing it.
I believe it is essential that we teach our students about the role of unanswered questions in driving human knowledge. The three great geometric problems of antiquity were not solvable with compass and straightedge — though the Greeks and many later investigators devised other tools and concepts and used them to solve the problems. The work they did under those limited conditions produced many prodigious mathematical achievements that, nevertheless, were not solutions to the original problems. A quick look at the careers of the great 20th century mathematicians shows that this is still what drives the science of mathematics forward today.
This section wraps up with another list of great mathematicians of the Roman imperial period, still called Greek, remember, because Greek was the language of intellectual and financial transactions:
- Ptolemy (120 A.D.) —
“He wrote on many subjects, from astronomy and geography to astrology, but his most famous work is the Syntaxis, known today by the nickname given to it by Arabic scholars many centuries later: Almagest, meaning “the greatest.” Ptolemy’s book provides a workable practical description of astronomical phenomena. It was the basis of almost all positional astronomy until the work of Copernicus in the 15th century.”
- Diophantus (circa 220 A.D.) —
“Diophantus … was probably one of the most original of the Greek mathematicians. His Arithmetica contains no geometry and no diagrams, focusing instead on solving algebraic problems; it is simply a list of problems and solutions. In the problems, Diophantus used a notation for the unknown and its powers that hints at algebraic notation developed a thousand years later in Europe…. Diophantus usually worked out the conditions under which his problems are solvable, thereby confirming that he was trying to find general solutions.”
- Pappus (circa 350 A.D.) —
“Perhaps the most important part of Pappus’s work, from a historical point of view, was his discussion of ‘the method of analysis.’ …. Pappus’s discussion of analysis is not very specific. This vagueness ended up being very important, because the mathematicians of the Renaissance understood him to mean that there was a secret method behind much of Greek mathematics. Their attempts to figure out what this method was led to many new ideas and discoveries in the 16th and 17th centuries.”
Note that, once again, the presence of an unsolved mystery drives mathematical discovery and innovation far more effectively than neat and clear presentations of methods and solutions. I think I’ll put that on a poster in my room this year, to answer all the complaints I will get along the lines of “Why don’t you just tell us how to do it?”, as though driving through difficulties to solutions and then discussing their merits has no intrinsic value.
- Heron (10 A.D. – 70 A.D.) is out of chronological order because he is noted for trying to bridge the chasm between the completely abstract endeavors of “scientific” mathematicians in the Greek tradition — the others named here — with the practical math that obviously was also developing throughout that time. Heron was more of an engineer, but his namesake formula for the area of a triangle in terms of its side lengths remains in the modern curriculum.
The authors note that both the “scientific” mathematical tradition of the great heroes of math and the “sub-scientific” mathematical tradition of merchants, government officials, mariners, and soldiers featured recreational problems, challenge problems, and puzzles. Like the Babylonians, those at all levels of the Greek mathematical tradition liked to do math “off the clock” as well as on. In fairness,we should note that we don’t have any indication of how common that was with the Egyptians, as we have spent much more time digging in their burial grounds than in the hearts of their cities (probably because those cities are under modern cities).
Well, that’s as far as I’ve gone, but stay tuned: this is just a first installment. There will be more of this, especially when I encounter historical information that I know I’ll use to help my troops believe in the mission in my classroom.