Market Scale Puzzle - Reflections

 I was not able to solve this puzzle. I’ll first discuss my attempts with the problem, an interesting pattern about the solution, and finally some comments about how I plan to use this problem.

One of the biggest pit-falls I ran into was realizing that the weights did not necessarily need to sum to 40. However, even after this realization, I worked with this assumption for a while without success. One of my closest early solutions was <20, 15, 3, 2>. With these, one can measure most weights, however any number containing 9 is an issue. After this near success, I searched for solutions which had some small numbers and one big number. Let this be Strategy 1.

I realized at one point that if I was able to find 3 weights which can measure everything up until 20, then I could make the 4^th be 20 and the problem would be solved. I was unable to do this, but it seemed like a promising line of thought initially. Let this be Strategy 2.  

Before I gave up, I was testing ‘mid-sized consecutive integers’, for instance <11, 12, 13, 14>. However, with these there is no way to sum to 40. Let this be Strategy 3.

The solution of 1, 3, 9, 27 is very clever. In a way, it is a more nuanced version of ‘Strategy 2’. Both the solution and this strategy use one large weight, but the solution leverages it to weigh sizes in the mid-range, not just larger numbers. In general, the only strategy I can imagine someone coming up with is realizing that the weights 1 – 40 must be divided up into 3 groups. The first group measured with small numbers, the second group being the difference of the large weight with the weights of the first group, and the third group being the sum of the large weight with the weights of the first group.

To understand the solution better, I curated the following table on excel. By color coating whether a weight was on the scale, as well as which side of the scale it was on, a very clear patter emerges. The weights are periodically taken on and off the scale as the target weight increases. The larger the weight, the longer the period. Noting that each weight can be written as 3ⁿ, n = 0, 1, 2, 3 , we see that the period of some weight 3ⁿ is 3ⁿ⁺¹. Noticing this pattern makes it easy to generate a table describing which combination of weights one would need to measure any weight from 1 – 121, provided one has access to weights sized for 1, 3, 9, 27, and 81.

This is a really excellent ‘low floor, high ceiling’ question. I intended to use this in the first week of the more advanced classes I teach in order to help students become familiar with whiteboards. A very simple extension of this question might be:

How many weights do you need to measure every weight from 1 – 200? What if I give you some random number N – how many weights do you need?

I might make this question DRAMTICALLY easier by first asking them to solve the case using 3 weights from 1 – 13. A less revealing hint would be to tell them that the 4 weights will sum to 40.

Word Problems in My Own Teaching

 Although I’ve come to appreciate the value of word problems in a historic context, I can’t help but feel that much of their original purpose is obsolete in the modern age, particularly with the normalization of algebra. In our age, I feel the main value these problems offer is the translation of stories to algebra. This type of modelling is a ‘real-world’ skill, even if the quantities described are unrealistic. However, because these word problems have the opportunity to tell stories, I believe the types of word problems we choose to implement should focus more heavily on student engagement. Although we might hope for students to let their imaginations run wild (as suggested in the comic on the blog), the truth is many students will scan questions, extract the needed information, and ignore the story. I believe the story could serve a much higher purpose.

In my classroom, I hope to have students engage with word problems by offering them choice. I see this idea reflected in ‘fair share’ questions – here, the student first chooses and/or motivates the question and model. This element of choice shows them how they might apply math in the real world – particularly, how they may need to translate their own ideas into an algebraic model. It is in this way that I hope to use word problems.

As a final note, I don’t see how I will make direct use of more classic word problems. Perhaps they will serve as source material as I look to develop my own, choice-oriented questions. Perhaps I will incorporate them as a historic novelty while trying to give context for problems. Regardless, I do not expect that classic word problems, out of context, will have a place in my classroom.

Ancient Egyptian Surveying

 As with everything Egyptian, I am shocked by the combination of how sophisticated their society was and how ignorant we are of their technologies. This article further revealed to me how deeply fundamental the Nile was to Egyptian technology – it is suggested here that it was the needs arising from the Nile’s unpredictable flooding that birthed some of the first formal methods of geometry (by needs, I am referring to the measuring and redistributing of lands)! The practical, authentic, understandable quality of these stories make them ripe for inspiring future students.

I am very impressed by the idea of the remen – it gives me further appreciation for the ratio that we so often take for granted. After some reflection, I’m a little surprised we don’t have our own special notation for √2. This is a concept that most modern people aren’t overly comfortable with, though I’m willing to bet that everyone in Egypt was fully capable of explaining the special meaning of a remen. Also, given that so much of Greek geometry was birthed from Egypt, I am curious as to how this unit was considered in the context of doubling the cube. Surely, they must have realized that the secrets to this problem were related to this common ratio.

Lastly, I have curiosity surrounding the ropes and knots used to measure long distances. The method of knot making must have been extremely precise – it is not a trivial question to determine how the length of a rope changes when you tie a knot in it. As such, they would have had to have a method for this that aligns precisely with the cubit. I would like to know how this was accomplished.

Babylonian Table of 45

Examples of the how the Babylonians would have used base 60 to create a multiplication table for 45

12        3, 45

15,        3

16        2, 48, 45

27        1, 40

40        1, 7, 30

Reflection - A Man Left Albuquerque Heading East: Chapter 7

 In this excerpt, we consider why, despite enormous differences in time and space, the mathematical content of some Babylonian word problems remain the same in modern times

Drawing from personal experience writing test questions, I can affirm the temptation to create problems as a function of the variables used in some given method. In order to test instrumental understanding, one intuitively provides variations of essentially the same question by varying which information is given. This often does not align with any practical situation, but instead forces the student to consider all possible ways in which some method might be used. In particular, by providing the student with information that would typically be the target quantity, they are forced to calculate some quantity that would in reality be measured directly. 

When all variations of some problem are considered together, they constitute a strong understanding of the method. This aligns with Høyrup’s observations that although many problems are presented with practical quantities, they are not the quantities one might have access to in reality. Lastly, without access to algebra, I personally can’t conceive of any alternate way of demonstrating some general method without direct application.

As such, I am (currently) of the opinion that the content of word problems has remained consistent over thousands of years because the methods themselves have also remained constant.

Babylonian Base 60 and the Visualization of Time

As noted in an earlier post, I speculated that base 60 was selected for its highly composite nature, as well as the ability for humans to use fingers and/or joints to intuitively count to 60. The articles considered here both present the composite nature of 60 as a likely motivator, though not with certainty. I was previously of the belief that this system must have been chosen for some reason before it was used as a tool to measure time and angle. However, learning about how Egyptians divided the day and night into 12 for separate reasons, I am now of the opinion that observation, utility, and number system are likely birthed together. This is supported E. F. Robertson, who believes that base 60 was not chosen for any specific reason, but instead was adopted by the natural way humans count.

I was also struck by how long time was measured with unequal units, something which amplifies the novelty of our mechanical, hyper-rational time measurement. From our class discussion, the main variables in the visualization / perception in time were:

Linear or Circular – Do you see each year as a circle? A line? A calendar? A spreadsheet?

Repeating or non-repeating (Does each year have its own line/circle? Or is every year envisioned on the same line?

Equal or un-equal units – Do all parts of the year feel as long? Are they consistent with our mechanical measurements of time?

I’m sure there is value in understanding time in all ways. That being said, I am curious how our Western (linear, non-repeating, equal units) understanding of time compares with the intuition of people throughout history. 

Babylonian Use of Base 60

  I can conceive of two distinct reasons the Babylonians may have used base 60:

1.       Convenience when counting with hands. One can count up to 5 with one hand and then use the other to count groups of 6’s. In general, because our bodies offer several easy ways to count up to 5, counting in base 60 is aligned with the body’s natural ‘overflow’ number. Once we run out of things to count (fingers, toes, appendages…) we start again, leading naturally to base 60.

2.       60 is the smallest positive integer to have 12 factors. The highly composite nature of 60 makes it practical, as it can be broken into many kinds of smaller, intuitive portions. 

A bit of research suggests that point (2) above is the primary reason for the Babylonian use of base 60. It is noteworthy that although we’ve dropped base 60 from our number system, 60 is still found in many modern applications such as the measurement of time and angles.

Note: Accidently made this post on the wrong blog - shifting it here now

Crest of the Peacock - Chapter 1 Reflection

 I enjoyed this overview of the transmission of knowledge, though it was a bit difficult to process given my relatively poor understanding of history. That being said, a few things of note:

1.       Although I was aware that some (much?) of Greek mathematics originated in Mesopotamia and Egypt, I did not know that this knowledge came from Greek mathematicians travelling to these places and acquiring this knowledge first hand. This came as a surprise to me because of my relatively ‘fuzzy’ understanding of when these civilizations existed.

2.       Figure 1.3 which illustrates mathematical activity through the dark ages was extremely interesting – I had no idea that height of mathematics in China and India corresponded so closely with this period in European history. I suppose this felt ‘surprising’ to me because I was wrongly thinking of the entire planet as being in a dark age, rather than just Europe. I hope to learn more about the developments during this time – I feel that knowledge of this period would be very helpful in my own process of decolonization.

3.      Somewhat disappointing is the lack of inclusion of Amerindian mathematics, specifically of the indigenous people who lived in what is colonially known as Canada and the USA. Landon (1993) states that “Traditionally, our ancestors preferred to rely on the intimacy and interactive characteristics of the oral tradition as a teaching and storage medium for what they knew of numbers, order and pattern. When a great many elders died prematurely during the epidemics and wars that were part of the conquest of the Americas, we lost a significant portion of our mathematical traditions”.

This suggests that this omission is not for lack of desire, but rather lack of available content.

 

Landon (1993) – American Indian Mathematics - Traditions and Contributions

https://www.pps.net/cms/lib/OR01913224/Centricity/Domain/179/pdfs/be-ai-ma.pdf

Integrating the History of Mathematics in the Classroom - Discussion Post

Prior to reading this article, I had an uncertain belief that the history of mathematics could play a role in teaching math. I recall reading a history of Greek mathematics as an undergraduate – this book made me feel as though I was the one ‘discovering’ mathematics. I believe such experiences are more accessible in universities (relative to secondary schools) because this feeling of discovery goes hand-in-hand with the creative side of mathematics. To inspire mathematical creativity in high schools, I frequently see the inclusion of ‘math-games’. This inclusion ‘feels’ correct, however the mathematics often feels arbitrary and disconnected from the curriculum. I am curious if history can act as a bridge between curriculum and the more playful, creative side of mathematics.

While initially reading the article’s list of ‘Objections to the incorporation of the history of mathematics’, I noticed many of the same objections had been made in Skemp’s article explaining why teachers opt to teach instrumental rather than relational mathematics. In particular – Lack of time, difficulty with examination, lack of teach expertise, and overburdened syllabi making new incorporations difficult. I speculate that the incorporation of the history of mathematics will be essential in the transition from teaching instrumental mathematics to teaching relational mathematics.

I felt a connection with the assertion that the history of mathematics will allow students to see math as beautiful, creative, and dynamic, rather than a purely formal and rigid science. Particularly 7.4.6 (i) ‘Errors’ presents opportunities for students to see the natural and intuitive side of mathematics. I feel strongly that activities based around historical errors can help cultivate the feeling of discovery I previously mentioned.

I finished this reading feeling excited. I see concrete ways that incorporating history can enrich the learning experience, specifically in providing context, motivation, and creative inspiration for students. I have a stronger belief that we can use history cultivate a feeling of discovery and creativity in the classroom. Lastly, I am still uncertain as to how this type of information will ‘land’ for young mathematicians. I worry that those expecting purely instrumental understanding will be uninterested in (what may seem like) an arbitrary history.

Hello World

 I am very excited to learn about the culmination of mathematics in history <3