Thoughts on word problems - Blog Response #8

It's quite interesting how word problems, something that has existed for millenniums, are still being questioned of its effectiveness and refined by educators even to this day. When it comes to writing word problems, I would want it well written. But what doe it mean to be well written? I believe that the problem should be clear to understand so that a student's success is dependent on their understanding of the subject, and not be impeded by language. However, as educators, as we focus on making sure that the questions we present for students to solve are unambiguous, includes all the required facts, and mathematically profound, it's easy to forget to add the flavouring that make math interesting. At least that was something I realized. Even if the problem presents itself as a plausible real-life scenario, is it truly and genuinely relatable to the students reading them? 

I want to be able to design problems that aren't just translations of math problems and create scenarios where students develop an algorithm in their head that they follow to solve such word problems. Perhaps, word problems can adopt a different medium that doesn't just involving reading a paragraph with specific criteria and writing down the solution that suits it. I would suggest to creating word problems that are open-ended where students have an opportunity to present their ways of thinking and create conversations in the classroom.

Ancient Egyptian surveying - Article Response #7

After reading the article on Egyptian surveying, I was quite astonished by how peculiar their measurement system was. Although there's not doubt that the Egyptians were masters of architecture, I still can't wrap my head around the fact how they achieved such remarkable levels of measurements and alignments given their limited tools at the time. But now I have come to realize how important surveying was for Egyptians in order to construct monumental structures, such as pyramids and temples, as well as for agricultural purposes, like measuring and restoring land after the annual flooding of the Nile. 

A question I have is regarding the creation of the unit, cubit. Why did the Egyptians decide to choose on a measurement that is not fixed in length? I acknowledge that the cubit is quite flexible, but I wonder what made it so convenient to dividing it into 7 palms or 28 fingers? In addition judging the terms "royal" cubit and "short" cubit, it seems to indicate that certain types of measurements are associated with pharaohs and power. Or was it tied more towards religious or cosmic beliefs, as speculated by the article as well? 

Babylonian word problems - Response #6

The context of what a world problem should be has always been an interesting topic to discuss about. Two opposing arguments that commonly comes up is whether a word problem should test one's practical or theoretical abilities. Indeed, when we solve word problems appear to have real world applications, we may deem it as practical. However, just because we replace numbers and symbols with sentences and phrases, does that make it actually practical? I think that applications that are contrived are quite meaningless. It is the ones that have real implications that prove to be beneficial. We need to make sure that word problems are true applications and not simply translations of mathematical formula. The difficulty arises when we try to represent the physical word using limited ideas. Questions that are ill made often have unnecessary jargon. 

Most of us here have most likely dealt with mathematics in an abstract and theoretical setting. Some people often hate dealing with proofs, while others thrive in an application-free environment. Both are great but I do think the former tends to not suit well for most individuals. We also see all the time, students become lost and experience math anxiety which leads to them jamming numbers with random operators. But realistically, what kind of application would ever satisfy a student? Should we provide the Babylonian ways about legal practices in dividing land heritance to beneficiaries? This was one of the first motivators of applications in history! Or do they need to know how to compute compounded interest rates and understand its growth? I don't think that the value of a word problem is always measured by how relatable it is. It is also important to hold high standards for both literacy and numeracy, while also effectively bridging the gap between them. 

Babylonian-style base 60 multiplication table for 45 - Blog Response #5

Here are some pairings of numbers that multiply to 45 that utilize base 60 fractions:

$$2 \times 22,30 = 45$$ since $2 \times 22.5 = 45$ so we can represent $0.5$ as $\frac{30}{60}$.

$$4 \times 11,25$$ since $4 \times 11.25 =45$ so we can represent $0.25$ as $\frac{15}{60}$.  

$$6 \times 7,30 = 45$$ since $6 \times 7.5 = 45$ so we can represent $0.5$ as $\frac{30}{60}$.

$$8 \times 5,37,30 = 45$$ since $8 \times 5.625 = 45$ so we can represent $0.62$ as $\frac{37}{60}$ and $0.005$ as $\frac{5}{60^2}$.

$$12 \times 3,45$$ since $12 \times 3.75 =45$ so we can represent $0.75$ as $\frac{45}{60}$.

  

History of time calculations, base 60 and base 12 - Blog Response #4

The two articles show how historical practices in measuring time and developing numerical systems are intertwined. The division of time into days, hours, and minutes has connections to the natural world and has evolved over time. It is an accumulation of astronomical observations, cultural practices, and historical developments. It is quite interesting how numbers like these may appear as simple or natural divisions, but they represent a story of refinement. Although the Babylonians used a base-60 sexagesimal system, it still influences how we measure time today. In fact, their historical number system reflects an extremely sophisticated understanding of mathematics and has left a lasting legacy on how we approach calculations and measurements. The remnants of their civilization can still be seen when it comes to angles, circles and spheres, and of course, time! 

There seems to be different reasonings between the two articles when it comes to why a base-60 system was chosen. The first article simply suggests that it is because 60 is a convenient number to express multiple fractions. It also learns towards historical context and practical aspects of time keeping in a more general sense. The second article goes in more depth and provides various theories. It focuses more on the technical aspects of numerical systems and their mathematical implications. 

Why Base 60? - Blog Response #3

    When I compare the numbers 60 and 10, I see that 60 has more factors 10. It has many factors that include 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60, while 10 only has 1, 2, 5, 10. This means that 60 is a lot more flexible and effective when it comes to dividing it by smaller numbers. I also think that certain fractions like halves, thirds, quarters, fifths, and sixths can be easily represented with a system based off 60.  The most useful application of 60 that I use in my daily life is, of course, time! It is indeed an important number to help us convert between seconds, minutes and hours. I also enjoy listening to music set at 60 BPM because it tends to be relaxing and emotionally powerful. 

    After doing research, it does indeed seem that the Babylonians valued 60 for its highly composite nature. Without needing to resort complex fractions, they were able to effectively carry out trade, measurements, and astronomical calculations. To add onto the point regarding astronomy, the Babylonians divided the sky into 360 degrees, a multiple of 60, to track celestial movement and accurately create calendars.

The Crest of the Peacock introduction - Blog Response #2

     It is quite surprising to read about the adverse affects of colonization when it comes to overwriting history in a mathematical context. It is disheartening to hear that European ideologies dominated in the 19th century, resulting in the devaluation of colonized peoples contributions. We see the same occurrence even further back in history as the Eurocentric model credits Ancient Greece for the origins of math and not Egypt nor Mesopotamia. I like how the book goes into more depth and shows us gradual improvements of figures that holistically represent the trajectory of mathematical development. Although the later figures depict confusing arrows that point in various directions and even having multiple of them converge to a singular spot, it emphasizes how math and culture is intertwined through diverse eras of countries and civilizations. Math is a collaborative process and we should aim to recognize all those who have contributed. 

    I also find it interesting that there are sources and possible factual evidence that describe Pythagoras travelling around the world to places like India. It makes me wonder if there are documented cases of any other notable scholars at time decided set off on foot or sail to explore the world in search of knowledge. Something that also came to my knowledge in recent years is that math concepts like place-holder values, solving quadratic equations, and the infamous Pythagoras theorem have all made their appearance in various regions. I wouldn't have ever guessed that the word "algebra" is of Arabic origin if I never took the time to inquire about it. I hope that our society continues to attribute mathematical findings to their appropriate contributors and accept it in all its diverse forms!

Why teach math history? - Blog Response #1

    

    My initial opinions on how math history should be incorporated into teaching is to give students a tour of how historical and ancient mathematics were tackled by past mathematicians. Ever since the modernization and formalization of math, students will be inevitably introduced to rigorous theorems and definitions. There are many math concepts that I struggled to grasp but something that shocked and gave me a sense of validation is when I learned that these concepts were giving headaches to the very mathematicians that developed them. This is why I think implementing history is a great way to help students gain an intuitive understanding of hard concepts by putting them through the lens of a period where information was not widely available. In addition, I was always inclined to believe that learning math is more individual focused during my time in school. But in fact, history says quite the opposite. Collaboration is the catalyst that drove generations of mathematicians from all over the world to work together and uncover the mysteries of our world.

    After reading the article, I appreciate the fact that they touched on the concepts of dialectical learning. Indeed, history is like the naïve and confused kid in their first year of high school figuring out where their classes are and who their friends are. While mathematics is now the successful and mature adult who’s got their life figured out. The contrast between developmental and polished stages of work provides completeness and an appropriate beginning and end to a wonderful story. I also like how the article points out math as a cultural endeavor. To gain practical skills and maximize utility are not the only reasons to learn math. In fact, to say to oneself, “I want to study math because it seems cool,” is a simple yet excellent reason to learn math as well. Students are often stripped of their creativity once they get lost in memorizing definitions to score well in exams. In fact, there is sublimity in mathematics, and we often times forget that art is inherently embedded in math.

    The article has further strengthened my belief that teaching historical math is important to students. I wasn’t aware that articles that included integration methods and models like this existed since I was rarely exposed to it as a high school student. Just recently, a professor of mine brought up the Navier-Stokes equation during a lecture and said that whoever can solve it first will be granted one million dollars! We need to pose more thought-provoking to students and provide the necessary motivation that they deserve. Thanks to the article, I hope to challenge myself in the future and incorporate these ideas in my classroom.

JJ Kim EDCP 442 2024_09_04

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