Blog Post #6: Egyptian fractions problem and reflection

Using the regular way of multiply the fraction to the totals before the death of a horse. This is the result: 

By using the Ancient Egyptian Unit Fractions visual. We are better able to visualize the distribution of horses among the three people, as noted here:

 

Blog Post #5: Egyptian division and reflection

 

As a reflection, Egyptian division uses multiples of 2 to get the answer, in which using this tactic to teach kids that struggle with division is useful (as discussed in class). The reason why is because you can see what goes into the number very easily as all the options you have are laid out in front of you. 

Blog Post #4: Response to readings on Babylonian word problems and 'algebra'

Reading on 'Babylonian Word Problems'

After finishing the reading, I found that practicality was considered through the convenience of the students' knowledge to solve the problems-- problems are not made too difficult for students to solve. In reference to the text, Babylonian mathematics seemed to be about word problems that had real life applications for trades training purposes. However, some of these real life applications did not really apply to the training at times, because an agricultural field worker did not need to make complex calculations with the work that they are usually assigned. In terms of generality, sometimes the situation is fit to practice the mathematical tool at hand and it doesn't really apply to the real life task. Pure mathematics seem like a conceptual form of mathematics that does not serve any hands on purpose to solve real world problems. While, applied mathematics is used to solve real world problems. I agree with the text that "Even when Babylonian mathematics is 'pure' in substance, it remains 'applied' in form." The reason why I agree with this is because Babylonians may have come up with really complex problems for agricultural workers that won't be applied on a daily basis; but if a problem were to arise that disrupted their day-to-day business, they would have the necessary tools and logic to solve them. Maybe the worker lost some seeds and they need to know based on the size of the field and the yield they calculated, how much they lost. These ideas relate to the history of mathematics because word problems play a large role in determining mathematics and its applications in the real world-- and how math has developed over time as "real life" situations have evolved. 

Reading on 'algebra;

From the reading on Babylonian 'algebra' from the Crest of the Peacock, I think that one could state a general mathematical principle in a time before algebraic notation using word problems to explain variables in the form of known objects. Mathematics is about generalization and abstractions because the mathematical equations/formulas can be used to solve real world problems. Other than the use of word problems portraying a situation giving real life examples to variables, I can't imagine a good way to show general or abstract relationships without algebra. 

Class Activity: Table 45

 

Blog Post #3: Response to reading of Crest of the Peacock introduction

Something that I really enjoy about math history is how it could provide some pretty useful context on why a certain theorem was looked into and then discovered. In my post, I will highlight three takeaways that I found super interesting from “The Crest of the Peacock”. The first interesting fact that I found was that the “Babylonians had developed a place-value number system, knew different methods of solving quadratic equations and understood the relationship between the sides of the right-angled triangle”. The reason why I find this interesting is because these two inventions are vital to the foundations of mathematics and geometry. In my own learning, I have used both of these concepts from grade school all the way up to my fourth year of university! The second interesting takeaway that I got from this text is that the “Arabic text... (which maybe loosely translated as ‘calculation by Restoration and Reduction')... refers to the two main operations in solving equations: ‘reunion’, the transfer of negative terms from one side of the equation to the other and ‘reduction’, the merging of like terms on the same side into a single term". I find this discovery to be quite intriguing because this tells me that there was so much thought that was put into something that is a strategy used when solving basic algebraic problems; where you gather the like terms onto one side of the equal sign and try to solve for the variable. Lastly, the Mayas "had accurate estimates of the duration of solar, lunar and planetary movements". Without the invention of telescopes and transportation into space, it seems nearly impossible to accurately predict something beyond the scope of observation. 

Blog Post #2: Why base 60 in Babylonian mathematics?

Why would 60 be a convenient, significant or especially useful number to use as the base for a number notational system, in comparison to the number 10? From the top of my mind, base 60 has more factors than base 10, such as 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60; while 10 only has 1, 2, 5, and 10. With 60 being divisible by more factors than 10, it may be easier to split it up into smaller subsections. Smaller subsections could account for more accuracy in measurement of time, etc.

How do we use 60s in our own daily lives in Canada and other cultures? All over the world, we use 60's in time by measuring seconds per minute and minutes per hour. And as discussed in class, the year consists of nearly 360 days which is almost divisible by 60. In our education system, when we learn about geography in school, where the geological coordinates that we use are also in base 60. Another example of base 60 being used in other cultures, would be the Chinese. 

According to Wikipedia, "in the Chinese calendar, a sexagenary cycle in commonly used, in which days or years are named by positions in a sequence of 10 stems and in another sequence of ten stems and in another sequence of 12 branches. The same stem and branch repeat every 60 steps through this cycle." It seems that the Sexagesimal is used frequently to measure time, angles and geographic coordinates because base 60 is divisible by 3, where as 10 is not. So when trying to express 1/3 in decimal form in base 10, you would end up having a repeating decimal. However, in a Sexagesimal system, 1/3 would be 20/60 which is basically 0.2 in the Sexagesimal system (https://blogs.scientificamerican.com/roots-of-unity/the-joy-of-sexagesimal-floating-point-arithmetic/). 

Blog Post #1: Response to Arcavi et al: Why teach math history?

In my opinion, math history should be incorporated into my own math teaching because it can give more context to why a proof or theorem exists by exploring what problem the mathematician was trying to solve given their background knowledge. It could be incorporated by using the mathematician’s background history as an anecdotal introduction to the lesson or main concept. It can also be incorporated in textbooks as a side blurb or fun fact to add more context to the information presented to better the understanding of the reader.

The first thing that I question is how under the affective predisposition towards mathematics section it states that “mathematics is an evolving and human subject rather than a system of rigid truths” because I believe that historical context of mathematicians could provide a better understanding as to why concepts were developed but, mathematics is primarily based on rigid truths to ensure that a proof or theorem is true in order for it to be useful in solving problems. The second thing that I agree with is that under the same section, “Not getting discouraged by failure, mistakes, uncertainties or misunderstandings…. Have been the building blocks of the work” because I can resonate with that in my own attempts to solve math problems; I am constantly faced with making errors and learning from them in order to deepen my understanding of the subject. The last thing that I disagree with is that “History may be tortuous and confusing rather than enlightening” because if the historical context provided is confusing and tortuous, then I believe that the information provided is ineffective or out of scope of the subject being taught.

My ideas have been further developed on the many ways that the history of mathematics can be taught in a classroom from when it is appropriate to assign a presentation or individual worksheet work. I have also developed a more humanized perspective on mathematics after reading this article, as it emphasizes that mathematics is developed through learning from errors of oneself and others.