Market scale puzzle + Connections

 

The four weights have to be 1, 3, 9, and 27. I tried working it out from 1 to 9 using different numbers first, then I realized the first 3 weights could be 3 to the powers of 0, 1, and 2. Then I tried 3^3=27 and it worked out perfectly. I don't know why these 4 numbers worked, and there seemed to be a unique solution to each value, but it was so satisfying.

If it were a one-pan scale with 5 weights, base-2 numbers with exponents 0, 1, 2, 3, and 4 seem to work! 1 + 2 + 4 + 8 + 16 = 31.

Riddles like this are so fascinating, because it shows patterns in Number Theory, and the fact that there might have been actual uses of it in the ancient world. For students to understand this idea and problem, I think the process that I went through - trying out with small numbers, recognizing pattern, expanding on the pattern, then deductively explaining how it works using what they know (though I'm still stuck on the last step) - is a good approach to new problems. The uncertainty and trial-and-error process is an entry point.

I don't know much about number theory, but there is something about the base 3 coinciding with the 3 possibilities in this problem. In other words, every weight is either not used, or used once for either the same pan or the other pan (3 possibilities). If I represent any herb weight value as N, 

N = 1a + 3b + 9c + 27d, with a, b, c, d belonging to the set {-1, 0, 1}

The above representation reminds me of linear algebra. But again, I still haven't figured out why!!

How would(n't) I use word problem in my own teaching?

The class discussion on Monday challenged my initial thoughts about the need for applicability in word problems. Initially, I believed that prioritizing real-world relevance meant using problems drawn from current events, up-to-date statistics, and relatable daily challenges. However, the discussion shifted my perspective. Word problems can instead focus on using mathematical tools for modeling, recognizing the patterns and beauty within math, contextualizing formulas, and humanizing math as a discipline. After all, problems in real life often appear as narratives, not simply as equations to "solve for x." This realization has made it clear to me that abandoning word problems altogether is not an option. The real challenge is in how I approach them.

Take, for example, teaching the Pythagorean theorem. A word problem could involve a carpenter needing to cut a right-triangular tile to fit an exact missing piece—or it could be a playful, imaginative story with the same core concept. It could ask students to validate the theorem by testing it with other triangles or exploring whether it holds true with different shapes. Students could also put themselves in Pythagoras' shoes, exploring the patterns as he did to understand his reasoning and how he developed the theorem. Additionally, they could be challenged to apply the theorem creatively, perhaps by designing a birdhouse, art project, or game. They could even create their own word problems and challenge their peers. In other words, my definition of word problems has expanded to encompass a variety of approaches, from incorporating humor and storytelling to fostering creativity and hands-on application. This broader view would hopefully enhance student engagement and their overall attitude toward math and word problems.

Response to Barnard (2014)

    Two questions that arise from Barnard's (2014) insightful article on "Surveying in Ancient Egypt" are: (1) Since "the basis for the Ancient Egyptian unit of length was the cubit, the length from the elbow to the tip of the middle finger," whose forearm was used to establish the cubit? and (2) Was Eratosthenes's calculation of the Earth's circumference, which implied the Earth was spherical, considered a radical idea at the time? The first question intrigues me because the answer could reveal which person, class, or gender held influence in defining standards—similar to how the "foot" was once based on the length of an English king's foot rather than the population average. Given that the tombstone mural depict people from different social classes, including children, participating in surveying, I wonder who had the authority to have their arm served as the standard unit of measurement.

    The second question is similar in nature to the controversy sparked by Galileo Galilei's 17th-century discovery that the Earth orbits the Sun. Did Eratosthenes’s conclusion that the Earth was spherical, with a measurable circumference, challenge any dominant religious or cultural beliefs, such as a concept of heaven existing above a flat Earth?

    What surprised me most was the Ancient Egyptians' understanding of the irrational number √2, which they estimated as nearly equivalent to 19.5 fingers (again, whose fingers?). Despite lacking precise tools, they approximated a value remarkably close to what we now know as √2. This raises an interesting question: have humans always been drawn to geometric forms like the square, leading to a natural curiosity about the length of its diagonal?

Response to Gerofsky (2004)

    As I mentioned in class, Francis Su's book "Mathematics for Human Flourishing" has reshaped my view of word problems, which often feel disconnected from real-world applications. Su’s idea of calling them "world problems" resonates with me, emphasizing the need to make them more relevant and practical. However, I was surprised by Susan Gerofsky’s "A Man Left Albuquerque Heading East", where she suggests that this disconnect might trace back to Babylonian mathematics! It made me wonder: has math education always been this way, or is it only in modern times that we’ve placed such a strong emphasis on real-life applications? Are we critiquing word problems through a contemporary, pragmatic lens, even though they’ve been integral to math instruction for thousands of years?

    Gerofsky (2004) references the Danish Assyriologist Jens Høyrup, who suggests that the Babylonians “had no other discourse available other than that of practical problems to express abstract mathematical ideas.” This is fascinating because I had never thought of word problems as a medium for expressing abstract math concepts. It’s similar to describing a real-world scenario today, then using math to model and solve it. In a way, word problems solve the challenge of not having precise measures or tools, serving as a bridge to understanding abstract ideas.

    I also found these questions thought-provoking: "Are word problems used primarily to train students in the use of methods without necessarily providing an understanding of those methods? Are problems chosen simply to illustrate the "methods at hand"?" (Gerofsky, 2004) The focus on “method” here is key. If the goal of word problems is to understand the use of methods, then I’ve definitely gained a lot from them.

45 Times Table in Base 60

Response to Scientific American and Connor & Robertson's (2000) Articles on History of Time Calculations

    When I think about the relationships among concepts of time, I often visualize them as interconnected fractions. For example, if we represent a year as a bar, it can be divided into four equal segments, each corresponding to a season (spring, summer, fall, winter). Each season represents 1/4 of the year, with the months grouped as January to March, April to June, July to September, and October to December. However, when I lived in Vietnam, I thought of a year differently—split into two seasons: the dry season from September to February and the rainy season from March to August, where each season represented 1/2 of the year. Similarly, I picture hours and minutes on a clock, divided into four quarters of 15 minutes or seconds. This recurring image of time as neat fractions highlights how intuitive and functional base 12 and base 60 systems are compared to our typical base 10.

    I was intrigued to learn from a Scientific American article that the concept of fixed hours is relatively modern, and that hours varied in length in the past. I had often wondered—but never explored—why we divide the day into 24 equal hours. Why not 12 or 60? And how did people account for the long days and nights that vary with the seasons? At the time fixed-length hours were introduced, this must have seemed like a radical idea, especially in regions far from the equator where day and night lengths fluctuate greatly. Reading about the historical shift toward dividing days and nights into 12 equal parts felt intellectually satisfying to me. While it might seem artificial or "forced" to impose such regularity on something as variable as the length of daylight, this decision reflects humanity’s intrinsic drive for patterns and consistency. It’s a fascinating example of how we adapt natural phenomena to fit our calculation systems, supported by both astronomy and the practical needs of daily life.

Response to Chapter 1 of "The Crest of the Peacock" - Joseph (1991)

    In the first chapter of "The Crest of the Peacock", I was struck by how transportation and increased mobility played a crucial role in advancing global mathematical knowledge. The exchange of ideas between Europe, the Arab world, Egypt, Babylonia, India, and China was made possible through improved transportation, enabling knowledge to be shared, discussed, and built upon - laying the groundwork for future discoveries. Such acknowledgement is so important as a piece of historical evidence of collaboration, so that students can relate and understand the benefits of collaboration, idea exchange, group-thinking, and feedback in the study of mathematics.

    I also found it fascinating that the concept of the placeholder emerged independently in four different cultures at different times and places. The fact that the Maya, despite being geographically isolated from the rest of the world, developed a similar idea is particularly intriguing. The need for a numerical placeholder seems to reflect the growing complexity of these societies - whether it be population growth, livestock, currency, or trade. 

    Throughout the reading, I kept wondering whether uncolonized countries - those never colonized by Europeans - have retained their own intuitive and culturally embedded ways of doing math and understanding the world numerically. What or who initiated the standardization of mathematical concepts globally, so that mathematicians now speak a relatively universal language? This contrasts with many other fields, including parts of science, where such standardization is not as prevalent.

Why Base 60?

1. The number 60 is both convenient and significant due to its divisibility by many integers (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60). It's the basis for time measurement, with 60 minutes in an hour and 60 seconds in a minute. Additionally, 60 is approximately equal to two months, or 1/6 of a year. Compared to 10, 60 is a more highly composite number, further highlighting its utility.

2. Beyond its use in timekeeping, 60 also plays a role in the Chinese zodiac system, where a full cycle spans 60 years. This system pairs 10 "stems" with 12 "branches" in sequence, and it takes 60 years for the same pair to repeat. The significance of 60 here stems from being the least common multiple of 10 and 12. In geometry, 60 appears in a full circle, as a circle contains 360 degrees—a multiple of 60.

3. According to John Manders’ Blog, 60 is also the least common multiple of 5 and 12, numbers linked to the human hand, with 5 fingers and 12 knuckles (excluding the thumb). The Babylonians, observing that the moon completes 12 cycles around the Earth in a year, used this knowledge to estimate the year’s length as 360 days—based on 12 cycles of 30 days each. In modern times, we see 12 hours on a clock, and items like donuts and eggs are often sold by the dozen. Additionally, a full circle has 360 degrees, and each degree is subdivided into 60 minutes, with each minute containing 60 seconds. In geometry, the internal angle of an equilateral triangle is 60 degrees (180/3 = 60).

   This website includes interesting facts about number systems in different cultures and languages. While no mentioned culture uses a strict base-60 system, many number systems are based on integers divisible by 60. This makes 60 a key factor in many systems, even when it's just a multiplier away!

Response to Tzanakis & Arcavi (2000)

    Initially, I believed that incorporating math history into teaching was important for two reasons: to humanize math as a subject and to highlight the value of curiosity, trial and error, and pattern recognition in mathematical discovery. Math is not just for the "left-brained" - those commonly perceived as logical and analytical. (In fact, can we truly categorize people as left- or right-brained?) Math is not dry or rigit but dynamic and ever-evolving. Introducing math history into teaching helps "humanize" the subject by revealing that it was not "discovered' in a perfect, complete state. Throughout history, people across cultures have been thinking, observing, hypothesizing, arguing, proving, and disproving what we now accept as common knowledge. They began by noticing patterns in the world around them and asking questions - many of which remain unanswered today. As a teacher, learning about math history with the intention of weaving these stories into my lessons, I aim to empower students with the idea that they, too, can be mathematicians - by being inquisitive, determined, diligent, and willing to take risks, just like those who contributed to the mathematical knowledge we have today.

    Tzanakis and Arcavi (2000) have added to my initial thoughts the emphasis on multicultural appreciation. The math textbooks and curriculum we have access to in North America are predominantly rooted in a Western perspective. However, different cultures have developed unique approaches to mathematical problems, bringing new insights. For example, while we often learn Western navigation methods, Indigenous cultures have developed their own methods that have been passed down through generations. Acknowledging these alternative approaches and appreciating the richness of mathematical knowledge from diverse cultures can inspire and engage students, letting them feel seen and valued in their own cultural identities.

    One specific example that stood out to me was the multiplication trick for 7 times 8, which reactivated my interest in different methods for enhancing fluency and making math more engaging. I was so intrigued by why the trick worked that I ended up diving into a full investigation, which ended up being fun and satisfying. This reminded me of how important it is to embody and find joy in math concepts. Offering alternative approaches - like tricks or patterns - can make math more accessible and engaging for students, particularly those who struggle with more traditional methods like rote memorization of multiplication tables. The more diverse approaches a teacher offers, the more students they can reach.

    Tzanakis and Arcavi's (2000) article has further inspired me to integrate math history into instruction, making the case for it even more compelling and inclusive. The insights and examples they provide are thought-provoking and valuable for future reference.