The Market Scales Puzzle

 

It would be interesting to look more into the concepts behind why these problems work out the way they do. For example, the nice properties we get when working with base 2 and the reasons we have binary representations of numbers. We could also consider how one might generalize the problem by developing a general formula for n weights. This problem was another great example of the "low floor, high ceiling" math problems which can engage a wide range of learners. Definitely reminds me of material covered in my Number Theory course with regards to working in binary as well as in other bases. I also originally thought there might be a connection to modular arithmetic and equivalence classes but I don't believe that was the case. 

Thanks for reading,

Josh

Exit Slip #2

 Hello twins.

The following is a response to our in-class discussion on the use of word problems in mathematics education.

To increase student engagement, I would personalize word problems by incorporating topics relevant to their interests, such as social media or video games with quantitative aspects. Another approach discussed in class was using more visual mediums to bring word problems to life, such as interactive videos or physical props.

I'm also interested in exploring the storytelling aspect of word problems. I've had success with a TED playlist called "Think Like a Coder," where students watched an animated series centred around solving coding problems. Creating engaging storylines with plots, characters, and other narrative elements could make word problems more captivating.

Furthermore, there are untapped opportunities to integrate collaboration into word problems. For example, having students work in teams to tackle complex problems, similar to escape rooms where groups solve multi-step challenges together. Lastly, I believe students could find it engaging to explore ancient math word problems, like those we examined in class. I found these incredibly engaging and hope students would share my enthusiasm.

Thanks for reading,

Josh

Chapter 5

Hello twins.

The following is a response to an article on Ancient Egyptian surveying and its relevance to the Tomb of Menna wall painting discussed in class.

My first question is how the Ancient Egyptians approached calculating the circumference of the Earth. I understand that celestial observation played a key role, such as using the height of the sun. I find it fascinating that they were curious enough to attempt such a feat and want to know more about how they achieved such accuracy in their measurements.

Another question I have is about how they established horizontal planes. For example, when hanging a picture frame, I often use a level to ensure it's not slanted. I'm curious if they had a tool similar to the modern level that achieved comparable results.

One aspect that never fails to amaze me is how the Egyptians built such grand pyramids and temples that still stand today. Considering the scale of these projects and the time period in which they were completed, it truly a magnificent feat of human engineering. It would be an incredibly challenging project to attempt even today, and the fact that they were successful thousands of years ago leaves me utterly speechless.

Thanks for reading,

Josh

Egyptian Division Problem

 

Babylonian-style base 60 multiplication table for the number forty-five

 

Chapter 4

Hello twins.

The following is in reponse to a reading on Babylonian word problems by our wonderful educator Susan Gerofsky.

The comparison between Greek theoretical mathematics and Babylonian practical applications represents a divide in the field that still exists today. It is similar to the instrumental versus relational debate on which approach should emphasized in the curriculum. While some post-secondary institutions distinguish between pure math and applied math for majors, others offer a degree that combines both. At first glance applied mathematics would be more useful for students, especially those in other disciplines. Integrating more applications into the curriculum can provide content for students who often question why they have to learn math in the first place. I believe this is why UBC saves most proof-based classes for the 300-level taken by math or computer science majors. I have found that many students often question the relevance or practicality of these more abstract classes that they believe are reserved for aspiring mathematicians. However, there is a beautiful elegance to the subject that may be appreciated by some students.

I have seen how many high school students struggle with word problems in their math classes. The additional layer of parsing sentences for contextual clues increases the difficulty for those with a more instrumental understanding. However, I find that students often lack the necessary strategies for approaching such problems, such as drawing a visual representation of the problem or translating equations from written text. Perhaps a greater focus on the concrete steps for dealing with word problems could be beneficial. However, the way most math textbook chapters are set up is also problematic. Most chapter problem sets contain mostly instrumental calculations with 1-2 word problems at the end of the exercise list. Once students have moved through the many instrumental exercises, they often see the word problems as an unnecessary challenge meant for more gifted students. Thus they often skip or put little effort into these questions and miss out on a valuable tool to solidify their understanding. Perhaps if we did a better job of consistently integrating word problem strategies into our lessons, students might feel more confident in attempting them.

Thanks for reading,

Josh

Chapter 3

Hello twins.

The following is in response to two short articles on the ways we measure time, as well as my own perspective on viewing time.

During the speculative phase on the base 60 system, I proposed that the number of convenient factors and divisibility into halves, thirds and quarters were key reasons. It is unsatisfying to discover there is no definitive consensus on the reason why. The theory I found most interesting was regarding the Sumerians being the result of a base 12 civilization joining a base 5 civilization. There is something poetic about two societies combining their numerical systems to create a potentially more effective system. During reflection, I also considered that there must be a number smaller than 60 with similar desirable properties. Another takeaway was how impressive it was to create a system for all numbers using only two symbols. It is difficult to imagine the challenges they must have faced with inventing their own numerical system.

I was surprised to learn about seasonally varying hours although it makes intuitive sense. Time has become so precise today that to know a time system existed with such variance is suprising. I cannot imagine being the first generation to explore time and what life was like before time was understood. Time is often considered our most valuable resource in life and measuring time has a profound impact on how we conduct our lives. I personally see years as a sort of slide show consisting of 4 equally divided quarters. It starts in January and ends in December with 3 months for each season.  Unfortunately, Vancouver's seasonal schedule is completely misaligned with this way of thinking. I prioritize months over seasons since important schedules are often tied to months rather than seasons. I valued hearing the perspectives of my classmates as it was something I hadn't previously given much thought.

Thanks for reading,

Josh

Chapter 2

 Hello twins.

The following is a response to Chapter 1 of "The Crest of the Peacock", a book covering the non-European roots of mathematics.

My first observation was the sheer interconnectedness between civilizations across the globe that were involved in the spread of mathematics. I was shocked at how many significant cultural influences and contributions were left out in the traditional Eurocentric perspective. I believe it is important to recognize how many cultures across different periods played a role in math history. The cross-pollination of ideas was vital in advancing the discipline and demonstrated the power of these ancient collaborations.  The sharing and transfer of knowledge between civilizations is impressive considering one could not take a plane ride or make a phone call to communicate. I clearly underestimated how vital the transportation and exchange of ideas were to the development of modern mathematics.

I also find it fascinating to compare how certain mathematical principles have shared contrasts between cultures. While there are certain similarities, each civilization has come up with unique solutions to universal problems. For example, the Pythagorean theorem has been discovered in different forms across the globe. This reminds me of the long-standing debate on whether mathematics was invented or discovered. Today we often take math for granted without considering the rich history behind its discovery. I found this reading quite meaningful when considering how vast the roots of mathematics are in the history of human civilization. 

Thanks for reading,

Josh

Exit Slip #1

Hello twins.

The following response concerns our in-class discussion of Babylonian mathematics and their choice to use a base 60 place value system.

An initial guess would be that 60 has the property of being divisible by 2,3,4,5,10 etc. Particularly being able to easily divide 60 into halves, thirds, quarters, fifths and other partitions could make it easier to work with. This is seen by comparing with 10 which is less easily divided. However, 60 still retains the property of being a multiple of 10 which could be helpful. 

In modern times we use 60 in our measurement system for time. We recognize that there are 60 seconds in a minute and 60 minutes in an hour. We also have close to 360 days in a year and 360 degrees in a circle. It seems that 60 and 360 are often used in a cyclic manner.  Once we have elapsed 60 seconds or 60 minutes then we reset for the next cycle. We cycle through 360 degrees in a circle and reset for another rotation. I had not previously considered the reasoning behind the prevalence of 60.

From a brief internet search, it seems that 60 having many factors and being easily divisible could have played a role. Another option is a potential connection to the calendar system of months and days in a year. Our modern choices for using 60 in time divisions stem from the Babylonian sexagesimal system. There are also ties to the ancient geographical systems for longitude and latitude where circles were divided into 60 equal parts.

Thanks for reading,

Josh

Chapter 1

 Hello twins.

The following is in response to an article about incorporating math history into the classroom.

Most people likely see mathematics and history as being distinct unrelated subjects. I find it intriguing to consider how one might blend these two elements together. One approach I have used with my students is looking into the people whose names frequent our textbooks such as Fermat, Euler, and Fibonacci. I have had positive reactions from students who are curious to learn more about the person behind these famous results. I could apply this approach to other aspects by exploring how modern math concepts have evolved over time.  

One initial reaction is that providing the history behind different math topics can deepen student understanding and foster an appreciation for the subject. Often times the ideas in mathematics are presented to us as finished products and I would be curious to know more about the process which led to that result. I am aware that mathematical proofs are presented in higher level mathematics but I believe one can go beyond this. Exploring the motivation or phenomena behind certain topics can provide purpose and direction to the classroom. This could be a welcome change to the current practice where concepts often appear out of nowhere with no backstory. In another class, I recently learned about the relational versus instrumental approaches to teaching and understanding mathematics. Providing the backstory behind math concepts is aligned with the relational perspective and could help to enrich the student experience. Math textbooks are full of precise results that students simply take at face value and apply to their work. I agree that having small excerpts sprinkled throughout the textbook which describe the history behind these findings could improve the reader's engagement with the text. Mathematics can often come across as an exact subject that requires perfection with no room for mistakes. I know from personal experience that this slim margin for error puts pressure on you as a student. Integrating changes in perspectives, revisions of initial assumptions and other errors could prove to be beneficial. By showing that making mistakes and having doubts is a major part of the discipline, one can help put their students at ease.

Reading this article has sparked my curiosity for ways to incorporate math history into my future classroom. One immediate idea is to have students research and create posters for famous mathematicians, who they were and what they contributed to the field. I believe students could find something personally interesting while investigating such a unique group of individuals. Another option that stood out is exploring famous unsolved problems in mathematics or those with no solution. There is something captivating about an unsolved problem that could be interesting to explore together as a class.

Thanks for reading,

Josh

Prologue

Hello twins.

This is my first post today. More to come soon. 

Thanks for reading,

Josh