The Market Scales Puzzle

I did not end up solving this problem. Here is what I tried:

First, I looked at all the different ways 4 numbers could be placed on 4 scales, and it seems like there is only one way to make 40 unique numbers when you start with 4. 

Then I thought about Egyptian multiplication. I tried doubling to make 40 and the numbers that were used are 1, 2, 4, 8, 16, 32. Then I tried to find a combination of 4 of these that could be used. I knew 1 would have to be one of the numbers so I could make odd (I wrote even on my page but meant odd) numbers. And I knew 32 would have to be one so I could make large enough numbers. 32+16=48 which is larger than 40 so assuming I was right that each way of adding and subtracting the numbers (that doesn’t produce a negative number) needs to be 1 – 40, 16 could not be one of the numbers. So, I looked at all the ways 1, 2, 3, 4, and 32 could be used to make 1 – 40 to see if I could figure out what the final 4 numbers would be.

I wasn’t able to make 16 however with these 5 numbers and I did not find a obvious way that 4 numbers could make 1 – 4 using this.

Some ways I could have went wrong are that it seems that either I was wrong that there are only unique ways to combine 4 numbers or, I should not be using the doubles as the numbers. If it is true that there are only 40 ways to combine 4 numbers then the largest number would be a+b+c+d so a+b+c+d would need to equal 40 but there is no combination of 1 ,2 ,4, 8, 16 and 32 that includes 4 numbers and adds up to 40. Beyond this there are lots of combinations of these numbers that repeat (16-8=8 but you already have 8 on its own same with 8-4=4, 4-2=2, and 2-1=1). So either my guess that I should be using doubles was wrong or my logic in determining how 4 numbers a, b, c, d could be combined was wrong.

Despite this I did find this to be an interesting exercise, and I feel that I understand the Egyptian way of adding doubles to create any number better. Specifically adding the doubles to get 1 – 15 made it really clear how they are essentially binary.

Word Problems Reading Response

This article spends a lot of time discussing pure vs. applied math and how the form that questions are written in may not be enough to fully determine whether it is a pure or applied question. It seems that the reason Babylonians used questions that appeared to be about practical math to ask non practical or more abstract problems was because these word problems were how they expressed all math. However, the ancient Greeks were able to express their pure math problems using symbolic notation which we would consider to be a more “pure math” way of writing these problems. We however, often use word problems in classrooms even when the topic we are discussing is not an applied math topic despite the fact that we, unlike the Babylonians, do not express all math problems in this way.

I think one of the main motivations behind creating word problems that are phrased as if they are something that could happen in real life but, that are actually so removed from reality that they are simply pure math problems is that teachers are trying to prove that what is taught in a math class is relevant to their students. I am not sure if this is the best way to prove relevance. We often feel the need to tie everything that is taught in a math classroom to a real-life skill. In the case where what we are teaching is something students will actually be able to use and apply in there day to day lives I think it is an excellent idea to highlight this fact. However, when what we are teaching is less applicable to day-to-day life I think that the creation of word problems that attempt to tie the content to real life without actually addressing a situation that would ever actually happen is not useful. I feel it would be much better to be honest with students. If the only reason students are ever given for learning math is that they will need it for day-to-day life I think we are not letting them realize the full value of the subject. Not every student will develop an appreciation for the beauty of math, nor will every student want to study math beyond school. However, exposing them to the fact that math is beautiful and interesting all on its own without practical application and that it is something they may want to continue to take interest in once they leave school seems to me to be valuable. Even though not every student enjoys art we do not tell them that they will need it for day-to-day life, instead we tell them that creativity and beauty are important on their own. I think when we teach subjects in math that won’t be relevant to the majority of students beyond the interest, they invoke that we should be honest about that. Instead of trying to tie everything to practical applications we should try and highlight how math can be inherently valuable on its own. 

Did Mesopotamian scribes have algebra? Response

Reading about the difference between how ancient Babylonians communicated math compared to how we do now made me question what the value of our current, symbol heavy, system is. The main thing that I believe makes our current system useful is readability. So long as someone knows how notation and mathematical symbols are used, a problem that would be a dense word problem using rhetorical algebra can be simplified to a more readable and arguably more simple equation. However, the trade-off here is that as we move away from a rhetorical way of expressing math it can be easy to lose the context and motivation behind the math. I could however foresee a situation where this disadvantage could become useful. I think the loss of context could be made into an advantage in a teamwork context. For a more technically complicated project it could be useful to have someone who is able to solve the math once it has been symbolized but who doesn’t necessarily need to know the potentially technically complex context behind it.

That being said when I consider the differences between rhetorical algebra and our modern system that is heavily based on symbols from an educational lens, I find some problems with our modern system. The heavy use of symbols makes it easy for kids to memorize, not understand what they are supposed to do to solve a question. Because symbols can be almost entirely taken out of the original context of a problem it becomes easy for kids to focus on the steps they’ve memorized to complete a question and not on understanding what the question is about and how it relates to the math they are learning. In this way I can see how the ancient Babylonian system may have been better. Students who only ever worked with rhetorical algebra may have had a stronger understanding of the math they learnt. Despite this I still wouldn’t argue that a system that only uses rhetorical algebra is best. I think there are times when rhetorical algebra is preferable but also times where symbols are a powerful and useful tool.

History of Time Calculations Articles Response

The way I think of an hour geometrically is as a block. I’m not sure where this came from, but it could have been from timetables in school. This is not very closely related to the idea of 60 min
utes in an hour. However, my imagery for a day is much more closely related to the idea of 24 hours as I think of days as a line with a notch to differentiate between time before noon and time after noon. I don’t have a notch to differentiate between before and after midnight since I think of the line as fading into and out of existence to represent the times when I am asleep. Once again, I am not sure where this way of thinking about it originated for me. I don’t feel that my perception/visualization of time is that close to how ancient peoples may have visualized it. I think likely any similarities between how someone from our time and someone from back then would have visualized time would come from the similarities between a sundial and an analog clock. I think a lot of people both now and back then would think of days as circular. However, I have never been great at reading analog clocks and use digital clocks when I can so this similar visualization to what might have been done in the past is not how I visualize time. 

I found the discussion in the second article about the lack of 0 interesting. It made me wonder how Babylonians delt with recording time specifically in cases where it was exactly on the hour (1:00, 8:00, 12:00, etc.). In modern times if someone were to say that something happened at 12, we would interpret that as it happening at 12:00 so I guess Babylonians could have done the same without needing 0. However, it would be more complicated if I said something took 12 as it would be unclear if it took 12 seconds, minutes, or hours. This could be solved either if the context was clear or by writing out 12 hours, 12 minutes, and 12 seconds. Instead of just relying on the number. To me from my modern perspective it still feels imprecise and that it could lead to confusion. This is the same with many of the ways that the lack of 0 would have affected Babylonians. In most cases I can logically understand that context would have been fully sufficient for them to know the difference between, for example, 1 and 60. However, my mind is so used to and comfortable with the modern system of numbers that I find that no matter how many logical arguments there are for how Babylonians would have coped with the lack of 0 I still find it disconcerting. 

Babylonian table for 45

Col I	Col II	Col I	Col II	Col I	Col II
1	45	6	7, 30	11	---
2	22, 30	7	---	12	3, 45
3	15	8	5, 37 ,30	13	---
4	11, 15	9	5	14	---
5	9	10	4, 30	15	3

The spaces I marked as --- are spaces where you get the base 60/ Babylonian equivalent of a repeating decimal.

Crest of the Peacock Intro Response

While I read this, I was surprised by the amount of travel and sharing of mathematical knowledge that was happening in the ancient world. I was not surprised that many different cultures were doing different types of math all over, but I had thought of it more as different cultures discovering the principals of mathematics in different ways. If I had had to guess before reading this, I would have said that many people would have discovered mathematical principals independently and that they likely would all have different names for them and different ways of proving things. I underestimated how much knowledge would travel with trade and with the fluctuations and changes in empires and also how much knowledge would travel by people specifically traveling to centers of learning and sharing what they knew.

Another thing I had not considered before this is the idea of looking at where modern notation and number systems and words for terms come from. I had thought (briefly) about where mathematical concepts came from but not about where the tools, we use to communicate these concepts come from. It was interesting learning about where different words for concepts originated. 

The last thing that surprised me was that the Pythagorean theorem had been stated in many different ways all over the world. The idea of different concepts being found in different ways around the world is not necessarily surprising to me. However, I find it surprising in the case of the Pythagorean theorem for two reasons. The first is that I associate the Pythagorean theorem so strongly with the modern way that it is presented. I think that this is because it is an idea that you learn when you are relatively young and then use a lot, so I have a hard time thinking of this theorem being stated in any way other than the way I learnt it. The other reason I find it surprising is that I knew before reading this that Pythagoras was Greek, so I think of it as a Greek theorem. I normally don’t know much, if anything, about the people our theorems are named after so, given that Pythagoras being Greek was one of the few facts I knew about someone who had a theorem name after them I found it memorable. It made me think of the Pythagorean theorem as a distinctly Greek theorem.

Why Teach Math History? Article Response

Before reading this article, I thought of math history being used in education mostly as a way of giving context to what we are doing in math classes and why we are doing those things. I thought it might be a useful tool to show that there is more than one way of doing things in math. Also, I thought it could be helpful to show students how much history and study went into creating the concepts that they learn in class. I thought that showing that what they are learning in class is the results of hundreds of years of work and study could be used to validate the feelings of students who are finding math to be difficult.

While reading one thing that stuck out to me what when different types of historical problems were being discussed. The discussion of problems with clever or alternative solutions stuck out to me. Before starting the reading, when I thought about history as a tool to show that there is more than one way to do things in math, I thought of showing how different cultures developed mathematical ideas in different ways and of using this to abstractly convey the message that its ok to be experimental and try different methods in math. However, when problems with alternative or clever solutions were brought up, I realized that there is a more concretely useful way in which math history can help kids: by showing that there is more than one way to do things. I think that showing alternative solutions to problems could be hugely beneficial in a classroom particularly for students who may struggle with doing a question in the first way that you teach. I feel that showing that problems can be solved in ways we wouldn’t necessarily think of can impower students to use tools from different areas of math to solve their problems (ex. Solving a problem that may seem more algebraic in a geometric way). This would be good because it can allow students to potentially apply concepts in math that they had an easy time grasping to concepts they might be more challenging. It is also good because I think it would encourage students to view math as a series of interconnected topics and ideas instead of as isolated topics.

Another thing that stuck out to me was when it was stated that a way to include math history in math education was to discuss mistakes that had been made by mathematicians throughout history. When I thought of using history to validate the feelings of students who are finding math to be a challenging subject I thought of using it to say “Look how long it took to develop these ideas. It is completely reasonable that they are hard to learn.” I didn’t think of using it to say “Look even these mathematicians who’s names we still remember today made mistakes. Making mistakes is ok and doesn’t mean your bad at math.” I like the idea of having students look at historical mistakes and having them try and show why they are wrong. It would teach them mistakes happen and are normal while also having them develop their mathematical skills.

Hello World

 Hello!

This is my dog when she was a puppy :)