Ancient Egyptian: Multiplication + Division

77 divided by 6

1                6 

2                12

3                18

4                24

5                30

6                36

7                42

8                48

9                54

10                60

11                66

12                72

1/2                3

1/3                2

1/6                1

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Total : 12 + 1/2 + 1/3              72 + 3 + 2 = 77

Using 5 as a divisor and using the Ancient Egyptian method of division, it will not work when dividing 77.  Trying 7 as a divisor, it will also not work.  I hypothesis that prime number divisors are limited because of its factors.

Ancient Egyptians: Unit Fractions

1/2 + 1/3  + 1/12 = 11/12

6/12 + 4/12 + 1/12 = 11/12

Pat gets 6 horses, Chris gets 4, and Sam gets 1.

The man anticipated the death of the horse.  The children would still get their respective share of the horses that would all add up to 11 and not 12.

Unit fractions helps with visualizing parts and wholes.  When we see a whole as a sum of its parts, we can easily separate parts out (dividing a whole into parts, then taking however many units we need).

Response: A Man Left Albuquerque heading East

    Many word problems are not practical and most students won't find themselves needing to find the height and length of a rectangle with only its area.  However, I think a reason for the seemingly irrelevant word problems is to get students to think about the problems at all.  When students are given models of what types of questions are possible (and many not very possible in their life), they may go out into the world and find similar problems.  Students hopefully become more inquisitive about the world around them as they are work on finding the height of a building with the length of the shadow.

    While the numbers and dimensions could be blown out of proportion, the basic math concepts used in many word problems can be applied to other problems.  If not all, some of the methods used in one problem can be used in another.  When students see the same type of word problem with small numbers, the big numbers, they might create a generality between them.  In Babylonian times, some of the word problems were found to be relatable to the working life of a Babylonian scribe.

    Shifting numbers and equations out of abstraction can help students understand the use of math and think about their answers.  I often ask my students if the answer makes sense logically.  If they find that the height of the bear is 20 m, does that make sense- or is it 2 m?

Why base 60?

    After doing the base 60 activity last week, I found that 60 has many factors, 1, 2, 3, 4, 5, 6, 10, ..., that are common factors with many other numbers.  On the other hand, 10 only has 4 factors.  Perhaps, 60 was used because it had enough factors to relate to other numbers (it could be divided evenly by many numbers), and it was not a very large number with too many factors that can be hard to handle.  The use of 60s can be seen in seconds, minutes, and hours.  It may have some relation to the days in a year as well and is connected to the earth's rotation.  

    From research, I found that Babylonian math originated from the Sumerians, a culture in Mesopotamia.  Two groups made of the Sumerians where one group used a number system based on 5 and the other group based on 12.  From interacting, they changed their number systems based on 60 so that it was consistent.

Reference:

https://www.thoughtco.com/why-we-still-use-babylonian-mathematics-116679  

The Crest of the Peacock: Chapter 1 Reflection

  The book states that rulers in the past recognized that "to control the past is to master the present and thereby consolidate their power". I take "controlling the past" as becoming knowledgeable about history and the origins of things because it is impossible to change what happened in the past. However, it is possible to change our beliefs and expand our knowledge base about past events. This statement is used more contextually by the author, describing how rulers used their broad knowledge and experience to strengthen the power they have in the present. To have knowledge and experience from the past is to have tool and strategies that can be used in the present to further oneself in life, no matter the context. However, how accurate can our experiences be of what truly happened in the past?

    Another point is made by the author on the issue of "who 'makes' science and technology". No matter how much knowledge we attain from the past, if it's heavily-flawed or is exclusive of important viewpoints, how useful is it for our present experiences? The author questions if indigenous scientific and technological was self-sufficient in the pre-colonial period. That is interesting to consider since I usually think of history starting from a European perspective, while there was no history that happened before the Europeans. It is as if "all significant historical developments in science and technology" were attributed to Europe and the rest of the world only contributed an insignificant amount.

    I found it surprising to learn about what happened to the contributions of colonized people. Their work was depreciated to assert European dominance. For example, Egypt and Mesopotamia contributed to the advancement of math before the Greeks, but respect for ancient Egyptian science and civilization declined as seen written in late-1900s writings. This subjugation led to a Eurocentric model of math with Greece. We see how the work of one nation can be taken away as their own in a struggle for power. It is important to question where we get our knowledge and how the knowledge came to be.

Joseph, G. G. (1992). The crest of the peacock. Penguin Books.

Tzanakis & Arcavi: Why Teach Math History?

In my classroom, I would use math history as a way to vary my teaching, making it more interesting.  Stories and understanding origins make things stick.  It can frame mathematics as a useful tool that was used to discover the things we know today. It also makes math less abstract or impossible to really understand as students analyze the development and sophistication of math language from its origins. 

I agree that the history of math is relevant to the teaching and learning process of both the teacher and the student. For the teacher, it can deepen passion and understanding thus helping teachers develop more interesting lesson plans and activities. I wonder about one objection to integrating math history because of the lack of assessment.  Is it necessary to be assessed?  I believe if the history is done in manageable chunks, it may be more easily digestible than math equations learned through instrumental or relational understanding.  History can be used as a tool to bridge language students are familiar with (probably English) and mathematical language which is largely foreign to students.  Students learn through stories and imagery, which most people have experience with growing up.

Some students are more art-geared (art, history, English), so looking at math through history may help motivate these students who are not very interested in math.  Math is often taught through instrumental understanding, so if teachers develop plans including history, maybe math could be better taught and learned.  I believe this can be done efficiently through some history of the origins of math and easier to integrate multiple media into lessons (videos, experiments (how math problems were solved back then, models, etc).