Response to “Did Mesopotamian scribes have algebra?” (Sep 23rd)

 The first point that made me stop was that the core mathematical idea can remain the same even when the notation are different. The Babylonians did not use modern symbols such as x,y, or x^2, but they could still solve equations by using words such as “length,” “breadth,” and “area.” Their methods for solving linear and quadratic equations were very alike to the methods we use today. So, mathematical symbols can be convenient or not, and we can also discuss whether a number system or notation is well designed. However, no matter which notation we use, it should serve the same mathematical logic.

That inspires me the way students learn mathematics now. I think students should also begin by focusing on the principle, instead of paying too much attention to mathematical symbols at the beginning. In my own teaching, I can use more words at first and let students explain what the quantities and relationships mean. After they have a general understanding, I can introduce the corresponding mathematical notation. This does not mean that symbols are not important. Standard notation is necessary because it makes mathematical ideas clearer and easier to communicate. However, if students only memorize symbols and procedures without understanding the meaning behind them, they may not really understand the mathematics.

Market scale puzzle

 

Response to “Babylonian Word Problems”

 The point that made me stop was the article’s discussion of Babylonian word problems and whether they were really practical. I have always thought that mathematics, in its earliest form, was closely connected with practical life. Mathematical ideas developed because people needed to deal with trade, measurement, engineering, accounting, or other economical reasons. However, this article shows that Babylonian mathematics was not only a tool for solving practical problems. Although many of the problems used situations such as grain piles or construction, some of them were clearly too simplified or unrealistic to be used directly in daily work. I think this shows that Babylonian mathematics was becoming more abstract. It was no longer only attached to practical production. It was also used to help students practice mathematical methods and see relationships between quantities from a mathematical aspect.

This also inspired me to think about modern applied word problems. In class, we discussed an example of a teacher who was trying to introduce the idea of proportion by using a problem about interior renovation. One student’s family worked in renovation, and the student had also participated in it before. Therefore, he knew that the proportions in the question could be affected by many different factors. He raised his hand and gave several examples to explain why the problem was not rigorous. If I were the teacher, I would not simply tell him that he was thinking too much and that it was only a simple math question. Instead, I would first praise the student's knowledge, and then explain that the problem was not designed by interior renovation expert to test real renovation knowledge. The question was designed to help students who do not know much about renovation see the proportional relationship between two quantities. I think this kind of answer can respect the student’s real-life knowledge, while also helping him understand why we sometimes use simplified models in mathematics.

time table

 

Response to the Readings on Time and Base 60

The first point that I noticed was the difference between the two articles in explaining why base 60 was used. I do not think there is a direct factual inconsistency. The Scientific American article explains that 60 is very convenient for expressing fractions, while the St Andrews article agrees with this mathematical advantage but questions whether it can explain the historical origin of the system. If divisibility were the only reason, then we might also ask why people did not choose 12 or 30 as a base. I think this is an important point for teaching. When we teach the history of math, we should tell students clearly whether we have solid evidence for a fact or only a well-supported explanation. More importantly, we should help them understand how mathematical ideas may develop through practical needs. The educational value is not only knowing who invented something or why a system was chosen, but learning how mathematics is connected to real human activities and why my students are learning math.

The second point that I noticed was that the Babylonian system could still work even though it had no true zero and no clear sexagesimal point. That reminds me of a problem in teaching. In many textbooks, we can find expressions saying that the limit of a function “equals DNE.” Strictly speaking, this is incorrect, because DNE is not a number. However, at the beginning, teachers sometimes need to balance students’ understanding with a fully rigorous definition. I think we should tell students that this notation is not formally correct, but we can temporarily allow them to write it when everyone understands what it means and there is no ambiguity. On the other hand, if a notation can cause ambiguity, it must be treated more seriously. For example, in trigonometry, if an angle is written without a degree symbol, it is normally understood to be measured in radians, and I must explain this clearly in class. Therefore, the Babylonian example makes me think that mathematical notation does not need to be perfect as long as it works, but teachers need to help students understand when an informal notation is acceptable and when precision is necessary.

Reflection on The Crest of the Peacock

One thing that surprised me after reading this article is that the history of mathematics can be very different depending on the purpose and perspective of the historian. Mathematics itself is based on proof, so usually there is a clear answer about whether it is correct or not. However, history is different. A historical explanation may not be false, but it can still be partial because people choose to focus on different evidence and different questions. The author’s criticism of the traditional Eurocentric history made me think that when we study math history, we should not treat one version as the complete truth. We need to keep an open mind.

The second thing that surprised me is how much attention can be given to the question of who invented a mathematical idea first. I do not deny that this question can have cultural or even political value. It is also meaningful to correct the underestimation of Chinese, Indian, or other mathematical traditions. However, personally I am not very interested in proving that one civilization was “first.” For example, ancient China had highly developed astronomy, calendars, architecture, and engineering. It is hard to imagine that these achievements could exist without substantial mathematical knowledge. To me, a more interesting question is how those mathematical ideas developed, what problems they were trying to solve, and why people needed them in the first place.

The third thing that surprised me was the network model of mathematical development. I think this model is more convincing than a simple line from Greece to Europe. At the same time, communication in the ancient world was difficult and expensive because of language and transportation barriers. Compared to the high cost of transmitting mathematical ideas across long distances, I believe developing some mathematical ideas independently could sometimes be more economical for each culture. Besides, it can also explain the development timeline of each culture better. However, I'm more interested in studying how math developed within each culture and what historical demands led to that development.

Reflections on the History of Mathematics

 

The first point that made me stop was that mathematics does not always develop because of practical needs. Many mathematical ideas were created to solve real or scientific problems, but some also came from logic or mathematical problems themselves. It reminds me the invention of complex numbers and non-Euclidean geometry. It means sometimes, math tools are not all necessarily designed for economical demands, mathematical development can also help us discover new things about the world.

The second point that made me stop was the difference between simply teaching historical facts and using history to guide how math class should be planned. Before this reading, I thought the history of math could be an interesting addition to my classes, mainly through stories about mathematicians, discoveries, and interesting old methods. Now I see that history can also help students understand what questions originally led to a mathematical idea and why the idea was needed. This can give teachers another way to introduce a topic instead of starting directly from definitions and formulas.

Hello world EDCP 442.

 Hello world. This is my first blog for EDCP 442.