Assignment 3 Reflections

 My final project for this class was completed with Jasmine and was focused on the history of the Abacus; I am grateful for the unexpected richness of this topic. When Jasmine first proposed it I was skeptical – I was under the impression that the Abacus served little modern purpose and was, at best, a novelty of the past. How wrong I was! The shift in opinion I experienced was one of the most radical of the entire semester! I am now of the opinion that the abacus can and should be used more frequently to help students develop strong arithmetic skills. In particular with Mental Abacus training, research strongly suggests benefits for children developing their mental math skills, or more generally, children being introduced to the arbitrary qualities of modern mathematics. Skeltzer’s paper does an exceptional job articulating the qualities inherent in the abacus which also pervade our modern number systems.

I had not considered the possibility that some of my peers might have trauma associated with mental abacus training. In hindsight, this is not surprising – such an effective method of teaching is bound to be abused by cultures which understand mathematics as a calculative speed test. Frankly, as I reflect on this, I’m not totally convinced that speedy calculative ability has any strong correlation with (creative?) mathematic ability… it’s as if we’re going to a spelling-bee in search of the next Shakespeare!

One of the most pressing issues in modern high school classrooms is the prevalence of students with relatively weak arithmetic skills who depend heavily on calculators for basic calculations. As a future teacher, I wonder how the Abacus might help me meet these students ‘where they’re at’. Although mental abacus training is unrealistic, there is the possibility that teaching students the basics of using physical abaci would help them to develop arithmetic skills in parallel with curricular skills. One might accomplish this simply by having a variety of abaci in one’s classroom. Much like a ‘Rubiks Cube’ or other physical math puzzles / objects, abaci might also provide students with something to keep their hands busy.

Lastly, this presentation was the first time I felt activity and lecture was correctly balanced! This has been challenging for me, so it was nice to see an appropriate balance manifest for my last presentation of the semester.  

Assignment 2 Reflections

This assignment was the first time I found myself trying to ‘walk the talk’ as it relates to what I’ve perceived to be the goal of this course, namely to leverage history to provide context and make topics more engaging. Initially, when I chose explore this history of compass and straight-edge constructions, I had expected to primarily focus on the Greeks. While researching the Greeks, I was reminded of the deeply philosophical relationship they had with geometry – although there were practical applications of geometry, the Greeks were moved by the spiritual and sacred elements that they saw infused with these shapes. In attempting to better understand this motivation, I found a paper titled: “The Ritual Origin of Geometry”. The paper speculates that the first desire to create ‘perfect’ circles and squares was theological in nature. It also drew my attention to the importance of ‘peg and cord’ constructions, of which compass and straight-edge were mimicking.

This paper helped me realize (to a degree) how to really make this topic engaging. I posit many students would find the history of ancient rituals a more interesting entry point than history of the Greeks. I personally am much more excited by these ideas and can hopefully channel some of this excitement to my students. Additionally, having students actually execute peg and cord constructions is a highly collaborative activity. Much like the dancing of Euclidian proofs, students have the opportunity to embody the math which is so abstracted on the page.

Frankly, keeping in mind that the purpose of math history (for me) is to inspire students’ interest, I feel that typical chronological ‘reporting’ of history does not serve us. We care why these things happened, not when. Taken to an extreme, this leads me to consider the morality of embellishing history so as to make mathematics more interesting and engaging. Is it immoral to lie about the origins of the quadratic formula for the sake of learning the quadratic formula? It is a question worth asking, though it may be relatively unimportant. As someone still largely ignorant of the histories, it may be that all mathematical origins are fascinating and can serve to captivate students without embellishment. That being said, if it turns out they aren’t, I find myself without a clear answer to this question.  

Link for UBC Students to downloads a copy of "The Ritual Origin of Geometry". I HIGHLY recommend!

https://link.springer.com/article/10.1007/BF00327767

Was Pythagoras Chinese? - Reflection

As math teachers, we have the privilege of teaching knowledge which was discovered and proven, independently, by several ancient civilizations. This means that for many of the concepts we teach, there are multiple unique histories that can be used to provide meaningful context. With this in mind, how should we choose what to share? To answer this, I want to lean on what I believe is the purpose of including history in the first place:

Integrating the history of math into our math classes serves to inspire students by providing context which explains why the concepts were originally of interest.

 With this in mind, teachers trying to decide which histories to share should firstly consider the ethnicity of the students being taught. For example, a class of 2^nd and 3^rd generation Chinese immigrants (not uncommon in Vancouver) would likely find the Chinese history associated with Right Angle Theory much more relevant (and perhaps even understandable?) than the Greek equivalent. Similarly, if one is teaching to a class with primarily European Ancestry, the associated Greek history would make more sense to teach for the same reason.

Regardless of which history one focuses on, it is important to communicate to the students that other ancient civilizations also developed this knowledge in their own, unique ways. It would be extremely colonial to present exclusively one culture’s history, particularly in a subject (such as Math) where the knowledge being taught was discovered, independently, by many cultures around the world.

Lastly, in Math, Science, Humanities, and any other subject, I think naming concepts, constants, or ideas after the original inventor/discoverer is a bad idea. I have two reasons for this:

1. Often, the assumption is that the person who invents/discovers something is the FIRST invent/discover the relevant idea. As is the case with Pythagoras, this is not always true.

2. It is a missed opportunity to make ideas more accessible. I generally believe that it would be better to have self-describing names, particularly for ideas which have been shown to be fundamental to the universe. Instead of Boltzmann Constant, why not Gas Temperature Constant? Instead of Planck Length, why not simply Smallest Theoretical Length (STL)? I suggest these names with a healthy does of sarcasm (I don’t know how accurate the are) but I believe their advantage is clear. In particular, they would remove some mystery and gatekeeping from the academic world.   

Euclid and Beauty - Reflection

 This past week I found myself wanting to incorporate ‘ancient problems’ as I developed lecture resources, namely so I could demonstrate the ancient nature of what was being discussed (geometric sequences). After some digging, I found question 79 in the Rhind Papyrus which I adopted in a modern context. This new-found desire to leverage ancient problems inspires me to take the information in ancient books (such as ‘Elements’) much more seriously.

Although I’ve read about Euclid and his Elements before, I now find myself much more interested in this material than ever before. Although the study of these books is a daunting task, I suspect it would be well worth my time so that I might more easily know where to search for source material as I continue to develop resources.

My partner and I agree that ‘Beauty is in the eye of the beholder’. With this in mind, I suspect that many (if not all?) people who see the beauty in ‘Elements’ are those already deeply familiar with mathematics. It is a generally accepted notion that simplicity and generality are highly valued by mathematicians. As such, I can only speculate on the impact Elements must have had at the time of its release - to see the world of mathematics derived from such simple postulates would have been peak intellectual experience. In the years to follow, those who would have learned with Elements would come to appreciate its simplicity. It is only after this appreciation that any student would see Beauty in this place .

Beauty is subjective and (I suppose) relates to what the subject values. As a drummer, I value complex rhythms – those who have not invested time in understanding rhythms and patterns would likely fail to see beauty in the same places as me. In this way, I’d argue that beauty is a relationship between two subjective places, one of which is probably a human.

Dishes Puzzle Solution + Reflection

 

When I solved this problem, I initially translated the given information into equations. Once it was clear I had 4 variables and 4 unknowns, I was confident I could solve it. With these sort of tools, this question was relatively trivial.

One alternate approach one might take that still involves algebra is ‘guess and check’. A solution might look like this:

(1) Make an initial guess for the number of guests.

(2) Calculate how many dishes would have been served for this number.(If you choose some number not divisible by both 4 and 3, you’ll quickly realize that this is a requirement of the problem.)

(3) Manually calculate how many dishes are required for this number of guests. If your guess results in more or less than 65 dishes, adjust your guess accordingly and repeat 2.

This approach uses a form of algebra, but doesn’t acknowledge that the number of guests can be written as a variable to be solved for.

I think that yes, it DOES make a difference as to whether or not we incorporate historical material from a variety of cultures. Math is a language / tool developed in (almost?) every culture to date – by favoring one particular culture, we do a disserve to both the students and the subject. I also believe there is inherent value in seeing problems that originate from your culture. Since we will likely be teaching in multi-cultural classrooms, it is natural that we might leverage historic problems from various cultures.  

Lastly, I feel that the imagery used in these questions contributes to the historic and cultural value of the questions. In this way, they may serve to better inspire and immerse students. Conversely, I suspect students who are NOT interested in the culture will simply try to parse out the needed information. With this in mind, historical problems such as this must be introduced strategically, ensuring they relate to the content already being taught.  

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.