Response to Gerofsky, "The History of the Word Problem Genre"

 I would wager to bet that outside of a few specialists and students of the history of mathematics, the response to finding out that mathematical word problems have a history as long as the history of human written records would be one of surprise and perhaps even incredulity. "Why would word problems be that important?" is the question I imagine many people would ask. It is also the question I found myself asking as I read the opening portion of Gerofsky's chapter.

First I must confess that though I was aware the Babylonians had a sophisticated mathematics, before this class, I was unaware that the word problem genre had been a part of it. When I learned that it was, my next assumption was exactly the one Gerofsky identifies early in the chapter: that Babylonian mathematics (including word problems) was "'merely practical' as opposed to later Greek abstract, theoretical mathematics. So the main "stop" for me in this reading was the insight of Eleanor Robson (as explained by Gerofsky) that the problems in Babylonian mathematics "functioned on two levels...On one hand, they taught practical skills and tested methods to future scribes; on the other, 'many of those methods no longer had real-life applications.'" As Gerofsky summarizes it, Robson's concept was of a Babylonian mathematics which was "at one potentially useful and obviously impractical."

This, it seems to me, could stand as a summary statement about mathematics as I have experienced it and intend to teach it. It reminds me that the point of mathematics is not to be "practical," though astonishing real-world applications have followed from many branches of mathematics, sometimes quite unexpectedly. When my 14-year-old daughter asks, as she does somewhat regularly, "When will I need to use this in real life?" my reply is straightforward: "Probably you won't. You should find other reasons to study math."

I answer in this way because I think - and it seems the Babylonians also thought - that there are benefits to doing mathematics that go beyond "practical" daily life application. I'm not sure how they would have explained it, but for me, math is a way of looking at the world, a way of stretching and training our minds - even a way of appreciating beauty. And it's fun, which I can't help but think was part of it all in Babylonian times too.

I recognize that one could debate whether "word problems" themselves are fun. Perhaps not as we usually encounter them in mathematics textbooks today. But part of the insight of Gerofsky's chapter is that for the Babylonians, such problems (often impractical) were the only means of discourse available for the expression of mathematical ideas. My question coming out of this reading is: what in our mathematical discourse is the equivalent? Perhaps emphasizing the abstract side of math has done some of our students a disservice. Is there a way we as mathematicians and teachers of math can use "word problems" or something similar to express deeper mathematical thinking, even to lead our students forward in their understanding, rather than merely "disguising" routine math in a way that leaves many students feeling tricked? Perhaps I can find out more by reading the rest of Gerofsky's chapter someday!

Reflection on Babylonian Algebra & Learning Algebra Today

 "One follows this method..."

So begins the explanation of the Babylonian solution to a pair of linear equations in two unknowns (Example 4.8 in our handout). What strikes me about the examples we considered in this handout are that they are steps - almost a sort of numerical recipe - for a given scenario. These steps are described not using symbols, but in words. And from what I can tell, it seems the aim in these problems is often to obtain a "standard form" from which familiar steps can be taken. That is, these examples seem to show ancient algorithms.

General mathematical principles are thus stated using concrete, narrative word problems that are then worked out using a sequence of precise steps. But rather than use any formulaic symbols, the Babylonian scribe worked with words and sometimes abbreviations. For example, to describe what we today call variables, the Babylonians used everyday physical dimensions. So, in the example 4.8 mentioned above, the unknows are represented by the Babylonian words for "length" (ush) and "width" (sag), while the product of these two unknowns is called "area" (asha). Thus, the language employed in the Babylonian problems, while corresponding conceptually in some ways to our use of these terms, is also different. One can, for example, subtract "length" from "area," and so on. 

One thing that strikes me is that even when a problem has nothing to do with land measurements, the problem is still framed as finding the "length" and "width" of a rectangle (or a field, perhaps). I have not thought a lot about how students start to learn algebra in grade 6-7 or so, but I would not be surprised to learn there are parallels to this approach. It would seem natural to have an intermediate step before arriving at full abstraction. Certainly before any abstract symbols are introduced, young students encounter word problems that describe a situation and ask for a numerical answer, but they are not manipulating abstract variables! 

Another things that seems to be happening in the Babylonian approach to solving these problems is a kind of "cut-and-paste" geometry. Rather than performing abstract operations on variables, the Babylonian approach seems to be the rearrangement of areas. For example, in Example 4.7, it seems they visualized a rectangle with an unknown length and width, then cut off the protruding "excess" length, slice it in half, and paste one half onto the bottom of the rectangle to form a perfect, larger square missing a corner. By calculating the area of that missing corner, they could figure out the dimensions of the entire shape. This was done by adding the "corner" to the existing area of the rectangle to get the total area, then working backwards to find the dimensions of the large square. Such a "physical" geometric approach would also be helpful for students learning algebra; simply the insight that there can be physical explanations for why certain mathematical processes "work" is encouraging to students who might struggle to "trust" the abstract representations.

The Babylonians have shown us that mathematics indeed is not all about abstraction and generalization. The examples we considered show us that math began as a practical, concrete tool that was taught through specific, concrete examples. The focus here is on the procedure to get a correct answer, rather than the universal law behind why the procedure worked -- but it is still mathematics, the mathematics of counting, measuring, and navigating the physical world. It is from such foundations that later abstract and structural mathematics could be developed.

A Babylonian-style Base 60 Multiplication Table for 45

 

How Do We Envision (and Calculate!) Time? - Some Reflections

 It seems likely to me that the way we come to think about time as children stays with us all through our lives, even if we later encounter other models for envisioning a year, or a season, or a month, or an hour. From my own childhood, my way of envisioning a year was really two semi-circles. Starting on the x-axis (not that I would have known anything about an x-axis at that age!), we are at the start of the spring. The direction of travel around the circle is clockwise. In the "height of summer" - a phrase I learned from a young age - we find ourselves at the top of the first semi-circle. From there back to the x-axis is the movement towards the fall. From fall, the journey through winter takes us along the "negative side" of the circle, with the "lowest" point of the circle falling somewhere around Christmas-January.

As it turns out, the circle was how I thought about everything to do with time. No doubt I was influenced (and my parents and teachers were influenced) by the circular clock and traditional watch faces. While I cannot think of any geometry I associated with individual seasons, the way I conceived of the months and days and hours was all the same: everything was a circle. That the months were also circles is interesting and perhaps is owing to the fact that my parents had a unique kind of calendar in our home that displayed each month in a circular way! So finding the 15th of the month always meant looking basically at the bottom of the circle to me (it was slightly different for 28, 29, 30, and 31 days, but it was always right around the bottom of the month). 

As I reflect on this conception of time in circular ways, I am struck that, as Lombardi explains, the Egyptians too developed their conception of time from circles. According to Lombardi, the Egyptians had a base 12 number system (a fact that contradicts O'Conner and Robertson's claim that "no major civilisation seems to have come up with [a base 12 system]"!) and so used a sundial (the circle!) divided into 12 parts to divide the interval between sunrise and sunset. But even in dividing the time of the night into 12 parts, as Lombardi explains, the Egyptians followed the movement of the stars in a "semicircular" way through the sky. 

Though circles seem to connect both the Eygptian and Babylonian contributions to our timekeeping conventions, it is a blending of the base 12 and base 60 systems that give us the full picture. I find it intriguing that all we have are theories for why these bases were developed, but I appreciate O'Conner and Robertson's proposal that the Sumerians (who preceded the Babylonians and developed the base 60 system) came about through a mixing of two peoples having 10 and 6 as their number bases. As they point out, this would account for the use of the "10 unit" in the Babylonian system as a remnant of the earlier decimal system. 

Two things I learned in reading the Lombardi article are: (1) that the meaning of the "hour" was not commonly understood to be a duration of 60 minutes for many centuries - "until the first mechanical clocks that displayed minutes appeared near the end of the 16th century, and (2) that the division of the circle of the clock into 60 "minutes" was originally terminology taken from the division by Greek astronomers of the 360 degrees of latitude and longitude into "minutes" and "second minutes," or "seconds."

Finally, I find myself reflecting on the significance of the circle as a way of conceiving of the passing of time. In my faith tradition of Christianity (not something I knew as a young child but came to later in life), the Celtic cross has always intrigued me: it is a traditional cross that features a circular ring intersecting the arms of the cross (it was developed in the 8th-9th centuries in Ireland and Britain). As I understand it, the vertical and horizontal components of the cross came to be understood as the intersection of the human and divine realms. But what fascinates me is that intersection is itself contained within the circle, likely a symbol of the sun, which itself represented the passing of time. For me, it is a reminder to me that we all seek to form our understanding of our place in the universe within the same "circle" of time.

Response to Joseph, "The Crest of the Peacock"

 As I mentioned in a previous post, while I had good math teachers, I struggle to remember even one time when a secondary math teachers made mention of the historical development of mathematics. So where did I get my ideas about math history? I ask this because what struck me as I read the introduction to The Crest of the Peacock is that I DID have ideas -- as it turns out, largely inaccurate ones! I guess you could say my views were very close to what Figure 1.1 of the text describes: "The 'classical' Eurocentric trajectory." I can only surmise that my sense of math history came from history courses I took or perhaps a course I once took in the history of religion and science. Or maybe I picked up concepts from reading short articles in popular science magazines over the years.

Whatever the case may be, the above explains why I found myself surprised on almost every page of this reading. I'll focus my thoughts on three such occasions:

1. While I was aware that the Egyptians and Mesopotamians had developed mathematics to some degree, I had no idea that, as Joseph says, "There is the full acknowledgement given by the ancient Greeks themselves of the intellectual debt they owed the Egyptians" (p. 5). I did not realize that Artistotle's teacher had studied in Egypt, or that Thales and Pythagoras reportedly travelled widely in Egypt and Mesopotamia and learnt math from these areas. In other words, I was unaware of the connections and transmission of math between different ancient cultures. Rather, I was under the assumption that the Greeks had developed their mathematics independently for some reason. How rich and interesting it is the learn otherwise!

2. Along similar lines, I was ignorant of the Arab contributions to the development of mathematics and indeed the sciences in general. The line that made me stop was this: "The course of European cultural history and the history of European thought are inseparably tied up with the activities of Arab scholars during the Middle Ages and their seminal contributions to mathematics, the natural sciences, medicine and philosophy" (pp. 6-7). I believe I thought Arabs had transmitted some information during the "Dark Ages," but I was unaware of anything like "seminal contributions." I didn't even know the word algebra was Arabic in origin! And the impressive list of Arab scholars in the footnote on page 7 also struck me. In fact, it made me upset: how could I have gone through high school and a 4-year liberal arts collegiate program and never encountered (so far as I can remember) anything about Arab scholarship during the Middle Ages? I thought the "Dark Ages" really were dark, not just in Europe, but everywhere...

It turns out the world of mathematical history is a much bigger and much more interesting place than I realized. I would have loved to learn some of this in my high school days.

3. It seems like a small thing, but I stopped at the line near the end in the section on the Maya where Joseph writes, "As regards zero, there are only two original instances of its modern use in a number system: by the Maya, and by the Indians around the beginning of the Christian era" (p. 22). Beside the interesting fact that that Maya were one of two groups who developed the concept of zero, I was puzzled by this statement. I would have thought zero was used in mathematics commonly in the ancient world and I found myself surprised to learn this was not the case. I don't understand how math works without zero, so I obviously have a lot to learn!

Which I guess I what I would say is the overall theme of my blog post: I have a lot to learn about math history! (This is motivation to make sure my students have a better framework than I was given in my high school days.) 

Response to Tzanakis and Arcavi, "Integrating history of mathematics in the classroom: an analytic survey"

Why teach math history? Before reading this chapter by Tzanakis and Arcavi, my views on whether, why, and how math history should be incorporated into my math teaching were largely unformed. I struggle to think of even one time in my high school math classes when math history was mentioned! Certainly it was never the focus of a class. Still, my view was that at least some attempt to incorporate math history should be attempted, primarily as an attempt to "humanize" math, providing connections to people and the world in which they lived. I would have said that providing some idea of when and where mathematical ideas developed would make the material more interesting to the students. But I would have struggled to make any meaningful suggestions regarding how this should be done.

There were a few "stopping points" for me in the chapter -- one of which was at the very beginning of the article. The authors write, "Mathematics is often regarded as a collection of axioms, theorems, and proofs" with the assumption that "the logical clarity of such a presentation may be sufficient for understanding mathematics" (p. 201). But, as they later write, in such a presentation, "questions and problem which constituted basic motivations for the development of an idea, as well as any doubts along the way, remain hidden" (p. 204). I was struck by how rarely I ever paused to consider the development of the math I was learning in high school (and university). This meant I missed out on seeing mathematics as an "evolving human intellectual process" (p. 207). The effect is that I learned to view math as something "outside" of this world, when the opposite is true.

Along these lines, I was struck as well by the authors' comments regarding mathematics as a cultural endeavor that was in many cases "developed for its own sake, motivated by aesthetic criteria, intellectual curiosity, challenge and pleasure" (p. 207). Understanding more about the history of math and the motivations of those who were part of its development would have motived me to care even more about it, and perhaps to ask different questions as a student. Maybe I would have become more curious as to the importance of various math topics in the times in which they were developed.

My views on whether and why to incorporate mathematics in my own teaching are not fundamentally changed after reading this chapter, but they are certainly strengthened and enriched. I am now armed with many more reasons for incorporating math history in my teaching. But what has changed even more is that I am beginning to form some ideas of how to do this. While some of the specific examples the authors mentioned in the second half of the chapter involved unfamiliar mathematical concepts to me, the concrete examples of how math history can be integrated in the classroom have given me a number of new avenues to pursue. I especially appreciated the ideas of historical packages, having students work to solve historical problems, and developing a play to "re-experience the life of mathematicians in the past, as a way to appreciate the human side of mathematical activity" (p. 229). 

Hello world!

 It's time to learn about math history...