Market Scales Problem

Babylonian times table for 45

 

Reflection on Babylonian Algebra

 General mathematical principles can be described with words rather than using symbols. I’ve read bits of Euclid’s Elements of Geometry, and I regularly had to adjust my reading to deal with Euclid’s style of explanation and argument, as it was written it what may be considered a more “rhetorical” approach. There was also less notation, and he didn’t use degree measurements, instead describing angles using fractions and multiples of right angles.

A big part of math is generalization and abstraction; however, it is not all of it. Being able to apply a general-situation tool to a specific situation is also important. For example, Babylonian mathematicians appear to have general strategies for solving types of polynomials but then require tables of values to substitute in for particular situations.

I mentioned how Euclid discussed geometric principles and proofs without much symbology. He started with a written description of building blocks (these “axioms” include the definition of a line, point, and plane) and then used these foundations to expand his ambition. 

I’m still shaky on the differences between rhetorical, syncopated, and pure algebra, particularly on the distinction between the first two. My current understanding is that syncopated is something of a transition stage between rhetorical and pure, where terms, written out in words, occupy the position of the unknown. When teaching story problems in an algebraic or pre-algebraic context, I would often describe an intermediate step between the story and the fully abstracted equation. I didn’t have a great name for it, sometimes calling it a “word equation” in contrast to the “story problem” and the “symbolic equation”. It would be a blend of symbols and words (for example, 2 oranges + 3 apples = $5.50), and some students appreciated this extra step. Perhaps this is like syncopated algebra and served a similar purpose: bridging the gap between words, symbols, and algebraic thinking.

Number Systems Articles Response

I have heard that that when we get older, time, in a sense, speeds up. We are confronted less often with novel experiences or big changes in lifestyle, and so the days and the years blend into each other much more neatly. For kids, life is messier, with huge changes in circumstance happening regularly and thus making time take longer. There’s an episode of the TV show Star Trek: Voyager in which the characters interact with a planet where time moves much more quickly. The people in the slower time appear perfectly still to those experiencing faster time. As teachers, we are working with where whose lives are moving much more slowly than ours. The philosopher Henri Bergson claimed that the passage of time depends partly on the individual, and that our inner lives play a key role in that experience. Considering the dramatic and chaotic inner lives of secondary school students, who knows how fast or slow they are traveling. 

I appreciated learning the history of second, particularly the part about how its length is now tied to a fundamental physical relationship. Scientists have redefined other key metrics in a similar way, including the kilogram (now based on Planck’s constant) and the meter (now measured using the speed of light). Moving these measurements away from definitions that may rely on the arbitrary whims of a dominant society to something more universal feels like progress. 

The author seems confident that the Mesopotamian base-60 system emerged from the combination of two other number systems. Is this scenario so much more plausible than Sumerian bureaucrats standardizing and codifying a number system from above? The author thinks so, dismissing the latter possibility out-of-hand. I’m curious whether we have evidence from elsewhere of two or more number systems combining to form a joint system. Could we model this scenario and see how easy it is for a person using one number base to interact and come to an understanding with another individual using a different base? Does it “naturally” allow for the combination of the two systems? Sumeria and its civilizational inheritors are among the first agricultural, sedentary societies. Do different societal needs, modes of production, and hierarchies lead to different number systems? It seems like agricultural societies would benefit a lot from a number system that allows for the easy creation of fractions, as it would make measuring and comparing amounts of grain or other commodities easier. 

Three interesting things in Joseph's The Crest of the Peacock

I enjoyed the figure on page 10 proposing an alternative map of the transmission of mathematical knowledge during the Middle Ages, particularly in how important the role individual cities played. Cities were often much more independent (culturally, politically, and intellectually) of the surrounding areas than they are today, so it makes sense they occupy such an important spot.

I had known that ancient and medieval India was partially responsible for our number system and that they made some incredible advances in mathematics, but I didn’t know the details, particularly that they helped spread trigonometry and indeterminate equations. India sitting somewhere in the middle of all these different regions and contributing to and drawing on knowledge throughout the world illustrates how much more connected peoples in the past were (at least at times) than moderns realize.

Civilizations growing comfortable with and accepting zero as a number is always fascinating to me. There was a similar resistance to irrational and imaginary numbers. I like to imagine Mayan and Indian mathematicians trying to convince a skeptical public that nothing is actual something. 

Initial thoughts on Tzanakis' and Arcavi's "Integrating history of mathematis in the classroom: an analytic survey"

In my math teaching, I have not had much opportunity to bring in the history of where tools and concepts were developed. Most uses of historical context would be superficial. Although any time to discuss Newton’s fascination with alchemy or Euler’s hat was enjoyable, I never thought it was a great use of time, particularly in quickly paced classes. I love history, and I would appreciate evidence supporting an increased use of it in high school math.

I appreciated the reminder of the messiness of math history. No matter where one falls on the argument about whether math is fundamental or discovered, progress has been characterized by steps forward, slides backward, and many detours. Students’ progress shouldn’t be dissimilar.

Weird or duplicative terminology or symbology can frustrate students. Historical context about why certain words or symbols are used rather than others can help students memorize them. Shared language among what appear to be different concepts sometimes can reveal unexpected connections.