Oct 2: Market scale puzzle

The four values of the four weights are 1, 3, 9, and 27. My strategy for solving this question included a combination of reducing the problem into smaller parts and guessing/checking. To satisfy herbs of weight 1g-4g, I reduced that 1g and 3g would suffice. Then, in order to satisfy 40 grams, I concluded that the remaining weights should add up to 36 grams. I began testing, starting at 5g and 31g for my third and fourth weight. As I encountered gaps, I would systematically increase the third weight and restart my testing process. Ultimately, I landed at 1, 3, 9 and 27 as the answer. I was shocked to notice that these were the powers of 3. 

Using the theme of powers, I immediately tested the powers of two for the second question. It also related to our previous decisions in the garden about Egyptian multiplication. We learned that any number may be represented using powers of two. Furthermore, 1+2+4+8+16=31 which met the criteria for the one scale problem. Thus, the five weights for the second problem are 1g, 2g, 4g, 8g, and 16g. 

I may extend the problem by getting students to explore how adding weights would increase the quantities of herbs we may weigh. Does this number go up at a constant rate? Exponential? Or something different? One may also have fractional weights consistent with the base system (i.e. ⅓, 1/9…) to examine how precise one may get with these quantities. Could we construct any fraction using base 3 fractions? Why or why not? Lastly, students may explore how the base changes when one adds or subtracts states of the question. 

Overall, this puzzle connects with my prior knowledge of binary numbers. Specifically, it relates to the fact that any number may be represented using a binary system. Since a third state is added to this puzzle (left, right, or not included), we may similarly use a base 3 system. 

Sept 25: Reflection on word problems

Word problems have become an element of math assessment that most students dislike. Why is that? First, the topics of word problems may be too dull. In hopes of making math more applicable and related to the students, word problems have heavily become based on real-life scenarios that, for the most part, have nothing to do with the interests or concerns of a teenager. Often, these problems, like calculating mortgages or mixing solutions, feel forced and uninspiring. Hence, students are unmotivated to tackle them. Therefore, I would incorporate more elements of storytelling and subjects that relate to teenage life so that word problems become an element of play in the lessons. That way, students can exercise their creativity in math. I would also stray away from using word problems in purely assessment settings. For example, word problems may be used as brain breaks or during free time as collaborative activities. This may relieve the pressures students feel when encountering word problems, thus generating more positive sentiments around them. 

Overall, my opinion of using word problems as a form of creativity and play stems from the ancient genre from 4500 years ago. As discussed, word problems may have been a method to test and push the understanding of ancient civilizations rather than strictly practical applications. They also may have been used in social settings as forms of debate, connection or discussion. From this idea, classes today should also incorporate word problems to push student imagination and engagement rather than purely real-life applications. 

Sept 25: Egyptian Division

 

Sept 23: Ancient Egyptian Surveying

The article on Egyptian Surveying unpacks some of the methods and units of measurement utilized by the Ancient Egyptians. It confirmed some of the observations we discussed in class regarding the mural. These included using the rope and knots as a measuring tool, and measuring fields for tax and reporting loss purposes. While similar methods are still used today, I was astonished at their units of measurement. Egyptians are known for their precise calculations and infrastructure. However, according to the article, “the basis of the ancient Egyptian math unit length was… the length from the elbow to the tip of the middle finger” (2). Since this measurement may vary from person to person, I suspect there may have been occasional variances. For example, if I were to measure an object using my arm as a cubit, this may be inconsistent to another individual's arm. Despite this potential irregularity, the Egyptians were able to construct intricate, precise structures such as their tombs and pyramids. 

Two questions arose as I read the article. The first was regarding their techniques on right angles. The article proclaims that there is no evidence of the Egyptians using the special cases of the Pythagoras theorem during their measurements. Instead, they may have used as set squares or rhomboids to construct right angles. As a result, I wonder how this approach would be applied in larger-scale projects. Did they have to build large squares for each right angle calculation? The second question was concerning the significance of astro-surveying in modern times. In ancient times, the use of the sun and stars was very influential in calculating components like orientation, angles for large structures, and time. This may be because of the lack of light pollution, which allowed celestial objects to act as useful tools. Consequently, I wonder how prominent these techniques are in modern constructions. Given the increase in light pollution and development of other technologies, how often is astro-surveying done today?

Sept 18: The genre of word problems

This week’s reading explores the history and purpose of unrealistic word problems in applied mathematics. After analyzing Babylonian tablets, the text suggests that the application of these absurd problems may stem from the challenges posed by realistic scenarios. Real-life problems often involved complex computations that were impractical to perform by hand in ancient times. Thus, they strategically used numbers that would calculate nicely. This reminds me of elementary math, where students also frequently encounter unrealistic problems. The primary goal in elementary math is to train students in computational methods, similar to the objectives of the Babylonians. Given this shared goal, I understand why this approach may have persisted over centuries.

I question if this is still the case for secondary math given the advancements in methods and technology. When students progress to secondary education, they gain access to external resources like calculators or the Internet. Hence, it makes it easier for them to complete calculations that are impossible by hand. Furthermore, as the purpose begins to derive away from practising methods to generate a deeper understanding, word problems may reflect more realistic scenarios. For instance, assessments in finance, trigonometry, and quadratics often incorporate practical, real-life word scenarios. However, I notice that students who have grown accustomed to unrealistic word problems in their earlier education sometimes continue to view word problems as useless and unimportant, even when they become more grounded in reality. How can we, as educators, reframe world problems so that it may change their perspectives? 

Sept 18: Babylonian-style base 60 multiplication table for the number forty-five

Above is my exploration of base 60 multiplication.

Sept 16: History of time calculations

During the modern era, most of the world perceives time as fixed, precise entities where all hours are consistently the same length. After reading the first article, I was surprised that this was not always the case. In the past, hours varied in duration, with summer being longer and winter being shorter. Personally, this made a lot of sense since the period of sunlight varied heavily during these seasons, especially for areas further away from the equator. This also reminds me of the common saying that the “days are longer” in the summer, despite the actual day remaining the same length. An issue I encounter with this idea, if it were to be applied today, is how society would apply this globally. Since the world has become vastly interconnected, it is difficult to allow regions to vary the length of their hours. This is because such constructs would make it extremely difficult for industries like global trade, travel, or anything that requires scheduling on a worldwide scale. The first article also staggered my perception of time when it discussed atomic time. Today, time is a concept that many do not question. From a young age, it is engrained universally that a minute has exactly 60 seconds. Thus, when the article stated that occasionally minutes may have an extra minute, it reminded me that even the most concrete findings can be proven inaccurate and that we, as teachers, must be open to accepting new ideas as they arise. 

Between the two articles, I note that the second one claims that no major civilization has “seemed to come up” with a base 12. However, from the first article, we conclude this statement is untrue. The first article argues that the Egyptians utilized a duodecimal system due to its countability on your hand and the lunar cycle. As a result, there exists an inconsistency between the two articles.

Sept 11: Response to the Crest of the Peacock

This week’s reading highlighted the presence of Eurocentric biases in math development and history. Initially, I was shocked to learn that many of the influential mathematicians who are accredited for the growth of math have instances of travelling or being exposed to math from different regions before coming to their conclusions. This exposure to varying methods and knowledge may have significantly influenced their learning and ability to conclude. Hence, I find it frustrating that these regions do not receive credit or recognition for the part they played in different European mathematicians' lives. Had this been done today, these mathematician would have lost their credibility due to their failure to disclose the sources and origins of the information they are presenting. Secondly, I am surprised that many did not find the mathematical skills of ancient Egyptians apparent. As a scholar, I would assume that ancient Egypt had a substantial understanding of algebra and geometry to build and sustain its complex infrastructures and societies. Hence, I wonder why more academics did not question the Eurocentric trajectory when it did not include Ancient Egypt. Lastly, I am surprised by the remarkableness of Mayan mathematics. Since it was an isolated region, one would expect the development of mathematical concepts to be done at a much slower rate. That is because these would lack the transmission of knowledge that Europe, Asia and Africa faced. Consequently, I wonder what other isolated societies were able to conclude similar findings as the Europeans without their exposure to Asia and Europe.

Sept 11: Why Base 60?

 Speculative phase:

Before researching, I speculated that 60 might be convenient for two reasons. The first 60 has a lot of factors. That is 60 may divide into groups of 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30. As a result, it is easy to apply in various applications. Secondly, base 60 is closely related to the measurement of space and time. The moon phases, sun and rotation of the earth are measured accurately using base 60. Thus, the division of time into 60 seconds for a minute and 60 minutes for an hour may be significant. Today, we use base 60 in various applications. The most obvious is in telling time. We also discussed in class that 60 is important in light and hertz. Base 60 also shows up in rotations and geometry in angles.

Research phase: 

Upon researching, I found three significances of 60 in the Babylonian number system. First, base 60 was influential due to its vast number of factors. Since it is divisible by 2,3,4,5 and 6, it easily applies to various counting problems. For example, counting and dividing crops like grain may be done using base 60. Secondly, base 60 was effective in astronomy and time. Other than the moon phases discussed earlier, base 60 accurately measured other celestial events and planetary movements due to angles being precise in base 60. As a result, it had applications in time telling and later became linked to mythologies around the cosmos. Lastly, base 60 had religious significance. The number was portrayed as divine due to its large number of divisors. It was also present in sacred geometry due to temple designs and rituals like circles. As stated, circles have 360 degrees, which may be expressed in base 60.

Response to why we teach math history

Before reading the article, I believed math history should be incorporated into the classroom since it can inspire and spark curiosity within students. Often, math can feel very fixed and laid out due to how it is taught during the early primary years. Its heavy emphasis on rules, theorems and computations makes the subject feel less humanistic and flexible. As a result, students fail to realize that math is a discovery that evolves throughout human history. Personally, I am an avid lover of films. Thus, the integration of films, videos or documentaries is one way I would incorporate math history into the classroom. Other methods may include projects, art representations, small-scale student-led presentations, and classroom discussions. 

While reading the article, I stopped at the statement “Lack of assessment” (203). This quote made me question why students will only value or pay attention to the material if it is being assessed. I wonder if other methods will promote student engagement with math history without having the added stressors of being graded. As a result, I become eager to read section 7.4 on “Ideas and examples for classroom implementation.” Upon reading this section, I was astonished at the various ways history could be presented. Particularly, I became inspired to introduce games, historical problems, and ancient methods of doing mathematics into a high school classroom. The introduction of unsolved and unsolvable problems may shock students who previously believed that math was an unalterable subject. They may also find relief in learning the existence of many methods to solve a problem. Lastly, I find that adolescents are eager to prove teachers wrong, and thus may be enticed by these problems.   

 After reading the article, I recognize that my initial methods for teaching math history were not seamlessly incorporated into my lessons. I previously believed that math history may only be presented as a separate unit, activity, or assignment. By exploring the different approaches to integration, I feel more confident in teaching math history in engaging ways and less constrained by the types of activities I can use.

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