Socrates and the Slave Boy

Throughout this conversation, Socrates leads a slave boy- who presumably has no formal training in mathematics- through a discussion about the area of a square, its dimensions, and the relationship between them. Rather than giving the boy any direct answers, Socrates asks a series of questions. Although the boy makes mistakes at first, he eventually arrives at the correct conclusion: that a square built on the diagonal of another square has double the area. Socrates refers to this process as "recollection," suggesting that the knowledge was already within the boy, just waiting to be drawn out. This made me think about how mathematical understanding is often built upon prior ideas. New concepts in math are not always completely foreign—they can often be uncovered through careful questioning and reasoning, which is exactly what the Socratic method is designed to do. In many ways, this approach still applies in math education today, where "good" teaching often involves asking the right questions to guide students toward discovering ideas for themselves, rather than simply memorizing formulas or procedures.

The Market Scale Puzzle

After trying a brute-force approach, I started exploring specific number patterns to find a more structured solution. First, I considered using consecutive even numbers like 2, 4, 6, and so on, thinking they might build up neatly. However, I quickly realized that even though they could add up to small values easily, they weren’t flexible enough to reach the larger numbers like 40 efficiently. Next, I turned to powers of 2—such as 2, 4, 8, and 16—remembering how the ancient Egyptians used them in their number system. While powers of 2 are powerful for addition-based systems, I found them limiting on a two-pan scale, where subtraction is just as important as addition. It was only when I began experimenting with powers of 3—specifically 1, 3, 9, and 27—that everything clicked. These weights, used with the idea that they can be placed on either side of the scale, allowed for a combination of positive and negative contributions. This meant I could weigh any amount from 1 to 40 grams, not just by adding up, but by balancing weights in a way that also takes subtraction into account. The powers of 3, it turns out, perfectly fit the logic of the balanced ternary system, which is ideal for this type of scale.

Questions and Surprises from Surveying Ancient Egypt

 At first glance, I wasn’t quite sure what to make of the image. My initial observations focused on the rope and what appeared to be the exchange of some kind of crop or grain. However, after reading the short paper, it became clearer that the ancient scene depicted people using a rope for measurement.

One detail that stood out to me was the unit called the remen, which I hadn’t encountered before. The remen is defined as half the diagonal of a square with sides of one royal cubit—remarkably close to the modern concept of a unit square with a hypotenuse of √2. I had never really considered how practically useful √2 could be in real-world applications until reading this. The ability to halve or double areas using proportions and ratios would have been incredibly valuable, especially for tasks like land distribution.

I had already suspected that celestial observations played a major role in measuring angles. As one of my classmates mentioned in class, it's fascinating how accurately ancient mathematicians were able to estimate the Earth's circumference using simple observations of shadows and basic tools like sundials, bays, merkhets, and plumb bobs (three of which I had never heard of before!). 

After doing a quick Google search to learn more about what a merkhet is, I started to wonder how interesting it would be to compare this ancient tool with the modern technologies we use today to observe the stars.

1) What are some modern tools or technologies for observing celestial bodies (such as telescopes, satellites, and space probes), and how do they improve upon- or differ from- ancient tools like the merkhet? 

Students could represent their findings in a variety of formats: a comparison chart, a presentation, or even a short play that dramatizes the evolution of these instruments over time!

2) How could a classroom activity be built around comparing an ancient tool (like the merkhet) with a modern one, to highlight both limitations and strengths?

One idea is to have students work in small groups to research both an ancient and a modern astronomical tool. They could then create visual presentations (such as posters or slides) that compare the tools across key categories: accuracy, materials, ease of use, and purpose. To deepen engagement, each group could simulate how their ancient tool was used (e.g., marking time or aligning with a star) and then demonstrate how a modern tool achieves the same goal- highlighting the evolution of technology and understanding.

More Thoughts on Babylonian Algebra and Word Problems

Reading through this excerpt reminded me that word problems originally stemmed from descriptions of operations, and I found it interesting that some of the examples given were geometric- aside from the first one involving barley. While not all Babylonian word problems were geometric in nature, this focus makes me wonder if such problems were especially common due to the inherently visual nature of shapes. Visual representations might have made abstract mathematical ideas more tangible, especially in an era without symbolic notation. It’s also possible that geometry played a larger role in daily life- such as in land measurement, construction, or astronomy- which could explain its prominence in early problem-solving contexts.

The notion of finding an unknown measurement and expressing generalized theorems was likely conveyed through words, in contrast to the symbolic notation used in modern mathematics. The question of whether mathematics is fundamentally about generalization and abstraction is a broad one. I believe there is mathematical beauty in both generalized theorems and abstract problems, as they reflect human creativity and intellectual depth. However, after exploring Babylonian mathematics, it's important to remember that many word problems were created for practical purposes. I presume these problems originated from real-world situations- such as food storage, construction, or taxation- and that a scribe may have decided that these recurring scenarios were worth recording on a tablet for future reference.

Thinking about how complex areas of mathematical knowledge were developed without the use of algebraic notation is fascinating. It raises the question of how general or abstract concepts were represented and communicated. Without symbolic algebra, early mathematicians likely relied on verbal descriptions, geometric representations, or practical scenarios to express relationships and unknowns. This suggests that abstraction in mathematics doesn't depend solely on modern symbols- it can also emerge through patterns, spatial reasoning, and logical structure, even in purely verbal or visual forms. That said, I also wonder whether the Babylonians simply didn’t explore some of the more complex mathematical ideas we take for granted today- perhaps not because of a lack of intelligence, but because such concepts may have been too difficult to express without symbolic tools. It's certainly difficult for me to imagine approaching higher-level problems without algebra, since its use now feels second nature.

Delving into the History and Autonomy of Word Problems

Two moments in this chapter stood out to me. The first was Høyrup’s (1994) observation that certain Babylonian mathematical word problems revealed a non-applied nature. When mathematical discourse is constructed and maintains a level of artificiality- such as the use of contextually strange quantities- it allows mathematics to achieve a kind of “autonomy.” This idea collides with my recent interest in the reconceptualization of curriculum through the lens of social justice and activism. Markwick and Reiss (2025) argue that individuals’ engagement with knowledge should foster personal development and contribute to greater social and cultural justice by promoting critical and proactive interaction with the social, political, and economic forces shaping our world.

This tension between autonomous mathematics and applied, justice-oriented education raises important questions: How can we use word problems to ensure that teaching remains socially responsive and meaningful, while also honouring the internal beauty and coherence of mathematics? Engaging students with historically- and even contemporarily-“autonomous” mathematical problems, which still appear in today’s textbooks, can prompt critical reflection on the cultural and historical forces that have shaped mathematical thought. At the same time, integrating applied, justice-driven problems can connect students to the real-world implications of quantitative reasoning. In this way, the curriculum becomes more than just a vehicle for transmitting mathematical knowledge- it becomes a platform for empowering students as critical thinkers and agents of change.

I continue to reflect on the role of word problems within the context of social justice. The second moment that deepened this reflection comes from one of the final questions posed in the chapter: Are word problems primarily designed to train students in the use of methods without necessarily providing an understanding of those methods? (Gerofsky, 2004). This question invites a critical reconsideration of the pedagogical purpose of word problems. When these problems are reduced to mechanical exercises focused solely on method application, they risk disengaging students from the deeper meanings, purposes, and implications of the mathematics they are learning. In contrast, when designed with intention and thoughtfulness, word problems have the potential to illuminate the social, political, and cultural contexts in which they are embedded. Rather than being autonomous or contextless, they could serve as entry points for inquiry, reflection, and meaningful dialogue- transforming mathematics into a tool for critical thinking and social awareness

Gerofksy, S. (2004). A Man Left Albuquerque Heading East. Peter Lang.

Markwick, A., & Reiss, M. J. (2025). Reconceptualising the school curriculum to address global challenges: Marrying aims‐based and ‘powerful knowledge’ approaches. The Curriculum Journal, 36(1), 1-14. https://doi.org/10.1002/curj.258

Deliberating on the Origins of Base 60 and Time

In the first article by Scientific American, Michael A. Lombardi (2007) reflects on the origins of the sexagesimal system, proposing that the base-60 system may have developed from observations of the positions and intervals of the sun, moon, and stars. In contrast, O'Connor and Robertson (2000), writing for MacTutor, present multiple theories- including Theon's divisor explanation, Neugebauer's weight system reasoning, and their own interpretation. They suggest that the Babylonians may not have chosen base-60 by looking to the heavens, but rather adopted it in a way similar to how we arrived at our base-10 system- by simply observing their hands.

While Lombardi's (2007) explanation feels more "scientific" in its astronomical reasoning, I find O'Connor and Robertson’s (2000) argument more intuitive. It seems likely that the need for a counting system emerged from everyday practical needs, long before anyone began carefully studying the movements of celestial bodies. This also makes me think about how deeply ingrained our understanding of "how to count" is. We’re accustomed to counting to ten using our fingers, but other cultures—such as those in China or the Middle East—have developed entirely different counting methods.

The idea that number systems could have originated from something as simple and universal as counting on one's hands makes the subject feel much more human. While we may think that counting to 60 using our finger joints is strange, perhaps we should also question our own entrenched assumptions about what “counts” as counting or mathematics. It reminds me that foundational mathematics is not only shaped by abstract reasoning, but also rooted in the tangible, lived experiences of people trying to make sense of the world around them. 

This could also be a fun and engaging classroom activity at the beginning of the year: simply asking students how they count. It would open up a space to explore the diversity of counting methods across cultures and encourages students to reflect on their own assumptions about numbers. From there, the class could dive into the sexagesimal system- perhaps by working with fractions or time measurements- as a way to show how different systems can be both practical and deeply rooted in history.  Activities like this can help students see math not just as a set of rules, but as a reflection of human creativity, culture, and  problem-solving across time.

Jagatia, A. (2021, September 3). How the way you count reveals more than you think. BBC Future. https://www.bbc.com/future/article/20210902-how-finger-counting-gives-away-your-nationality

Lombardi, M. A. (2007, March 3). Why is a minute divided up into 60 seconds, an hour into 60 minutes, and only 24 hours in a day? Scientific American. https://www.scientificamerican.com/article/experts-time-division-days-hours-minutes/

O’Connor, J. J., & Robertson, E. F. (2000, December). Babylonian numerals. MacTutor History of Mathematics. University of St Andrews. https://mathshistory.st-andrews.ac.uk/HistTopics/Babylonian_numerals/

Thoughts on The Crest of the Peacock

Some things really stood out to me in this introduction. I had taken a history and philosophy of mathematics course about six years ago, and that class - along with the professor (the late Thomas Fox) -  completely changed my previously one-dimensional perspective of math. It transformed how I saw the subject, filling it with the vibrant colours of culture, philosophy, and human story. 

1) In that course, we covered Babylonian mathematics, but it wasn’t until this week — through this chapter — that I learned "Babylonian" is actually an umbrella term. It encompasses various regions and civilizations such as the Sumerians, Assyrians, and Babylonians themselves. That realization really made me reflect on how history often gets simplified — sometimes to the point of distortion. It reminds me of the importance of being mindful about how we teach history: what gets emphasized, what gets left out, and whose voices are being centered — or erased.

I also think about my experience as a student in that class. Not once did I truly question the history or philosophies being taught. Ironically, Dr. Fox would often scold our class for being too complacent in our learning - especially when it came to mathematics. sMoving forward, I want to be more intentional with how I refer to "Babylonian" mathematics in my classroom, acknowledging that it encompasses more than just Babylon itself.

2) Figure 1.3, which presents an alternative trajectory for the so-called "Dark Ages," really struck me. Compared to Figure 1.1, it highlights just how much the contributions of non-European peoples have been minimized or erased in mainstream historical narratives. It made me think about the origins of our number system — the Hindu-Arabic numerals — and how much of mathematics we take for granted without acknowledging its deeply multicultural roots.

Mathematics, like every other subject we study, is a human endeavor. Yet colonialism, historical neglect, and deliberate erasure have shaped the way it’s remembered and taught - often giving disproportionate credit to white European men while ignoring or downplaying the achievements of others.

3) On the same note of cultural erasure and epistemicide, I’ve been reminded of what I know about Al-Khwarizmi and his immense contributions to mathematics -  especially algebra. When I think about what’s often referred to as “Arabic mathematics,” I’m reminded that it was shaped not just by intellectual innovation but also by displacement. Around 700 AD, conflict forced many mathematicians to flee to Constantinople (Byzantium, now Istanbul). It's a reminder that social and political violence have long disrupted knowledge production, forcing people to carry ideas across borders - or lose them entirely.

What I found particularly striking - and frankly, a bit depressing - is that Al-Khwarizmi himself acknowledged the Indian origins of the number system we now use. Yet over time, repeated translations of his work erased this attribution. This really highlights how translation is not a neutral act. What gets translated? What gets lost? Who gets credited? The process of translation can subtly (or not so subtly) reshape our understanding of math and its origins.

At the same time, I also find myself wondering: without translation, would we have anything at all? It’s a paradox. Translation has preserved so much, yet it has also distorted and erased. It makes me think more critically about how knowledge is passed down - and who controls that narrative.

Why Base 60? Why 60 then? Why 10 now?

 1) As mentioned by some in class on Monday, Sept 8th, the number 60 has many factors—especially when compared to our base-10 system.

• 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
• 10: 1, 2, 5, 10

It makes me think that using 60 as a base would be much more efficient for working with fractions. When calculating measurements for cooking, feeding animals, taxes, or budgeting materials, 60 seems to offer more convenience. Like Susan pointed out:

• 1/3 of 60 is 20, while 1/3 of 10 is 0.33 (repeating).

• 1/4 of 60 is 15, while 1/4 of 10 is 2.5.

It makes mathematical sense that the larger the base number (like 60 compared to 10), the more divisibility options you get, and the less you rely on decimals—though decimals are still possible. Sixty seems to lead to more terminating decimals overall.

2) Our calendar year today is 365 days, but in the past, it was often calculated as 360 days—a number divisible by 60. Similarly, we measure angles in 360 degrees, and in trigonometry, our "special triangles" often feature angles that are fractions of 60:

• 15 (¼ of 60), 30 (½ of 60), 45 (¾ of 60), and 60 itself.

Another obvious place we see 60 is in time:

• 60 seconds in a minute

• 60 minutes in an hour

But then there's 24 hours in a day. That part has always made me wonder. Why 24? Sure, it’s 2/5 of 60—but why not 30 or even 60 hours in a day? This probably ties back to the historical and astronomical roots of timekeeping. Ancient mathematicians would study the sun, moon, and stars (especially the North Star) to find consistent patterns. I suspect that’s also how they came up with the idea of a 360-day year.

The earliest systems of cosmology, especially among the Babylonians, used base-60. They invented basic trigonometry, produced detailed star maps, and could even predict eclipses with surprising accuracy.

3) A little research on Google also shows that the number 60 holds spiritual and cultural significance. In the Bible, it appears numerous times, and some followers interpret it's frequency in the bible as the point where people enter the final stage of life.

In the Chinese zodiac, 60 marks the completion of a full sexagenary cycle: 12 zodiac signs × 5 elements. For example, I was born in the Year of the Dragon, which is also associated with the metal element—a combo that only happens once every 60 years.

BibleStudy. (n.d) Meaning of numbers in the bible: The number 60. https://www.biblestudy.org/bibleref/meaning-of-numbers-in-bible/60.html

Reflecting on the Integration of Mathematical History and Teaching

1) Before this reading, I had been fortunate to take a History and Philosophy of Mathematics course during my undergraduate studies. That course deeply moved and inspired me to integrate historical perspectives into my own teaching. During my third field experience, teaching Grade 9 mathematics, I began the year by introducing Babylonian mathematics, the sexagesimal number system, Greek mathematics, and Pythagoras. Later, while teaching number sets, I briefly discussed the contributions of Cantor and Zermelo.

By the end of the placement, many of my students- whose cultural backgrounds included Greek, Arabic, Persian, and Chinese- expressed excitement and pride in learning that their cultures had played integral roles in the development of mathematics. However, I couldn’t help but question whether my integration of history had been too superficial or lacking in depth. I continue to reflect on my own perspective and how I interpret student progress. I often wonder: what would truly demonstrate to me that a student has meaningfully engaged with both the history and the subject of mathematics? And is such proof even necessary?

2) 

a) I really enjoyed reading about the genetic approach and the concept of the necessity of the subject. From my understanding, this approach is grounded in students' own developmental processes and emphasizes the why behind specific theories, methods, and concepts. This resonated with the idea that mathematical activity is not purely intrinsic but is shaped by cultural and social contexts. It prompted me to consider questions such as: Why were certain formulas created? What problems or needs prompted their development? What were the intentions behind them? As I reflected on these questions, another one naturally emerged: How can I foster an environment where my students are encouraged to ask the same kinds of questions?

This line of thinking reminded me of Francis Su’s (2020) chapter on exploration, in which he emphasizes that exploration is a deep human desire- and that mathematical exploration begins with questions. Tzanakis and Arcavi (2000) similarly note that mathematics is motivated by curiosity, challenge, and pleasure—aligning closely with the desire Su describes. For me, the why questions feel natural; they arise spontaneously, like popcorn, throughout the course of a lesson. Being able to flexibly adapt to these moments of curiosity and make space for them in the classroom feels vital to nurturing genuine mathematical thinking.

b) One point that stood out to me was an objection highlighted on page 203, where some argue that the absence of assessment could lead to a lack of perceived value for students, ultimately resulting in disengagement and reduced motivation to participate. However, the pressure and perceived role of assessment are highly context-dependent. In some cases, assessment may provide structure and motivation; in others, it might hinder curiosity and risk-taking. The tension between assessment and its perceived "necessity" in the classroom is not new, and it continues to raise important questions about how we define learning, success, and student engagement. When, then, is assessment truly necessary—and how can it be designed to support rather than undermine meaningful learning?

3) Some ideas I will continue to reflect on after reading this paper relate to the cultural and societal factors that have shaped the development of certain mathematical domains. In particular, recognizing the dominant Eurocentric narrative that is often implicit in the curriculum and subject material raises important questions: What would decolonizing mathematics look like? How can we broaden the mathematical story to meaningfully include diverse cultural contributions and perspectives? Which students’ experiences and cultures are excluded or marginalized in this history? And how can I bring those voices into the story?

References:

Su, F. (2020). Mathematics for human flourishing. Yale University Press. 

Tzanakis, C., & Arcavi, A. (2000). Integrating history of mathematics in the classroom: an analytic survey. In J. Fauvel., & J. van Maanen. (Eds.), History in mathematics education: the ICMI study (pp. 201-240). Dodrecht: Kluwer.  

Hello World!

    To whomever stumbles onto this blog, intentional or not, welcome! This blog will be a record of my (Anna Leung) encounters and interactions with insightful readings, exciting encounters, discoveries, and even clashes with the extensive knowledge waiting to be uncovered in this world we share. 

While I do not expect my visitors to show the same sentiment as the photo below, I do hope that this blog can serve as fertile ground for a seed for a love of mathematics to grow and thrive!