Fermat’s Last Theorem Film
I think that this film will make students realize that there are real actual people who research mathematics. They will also realize that it actually takes quite a bit of time to prove things in mathematics (just like how it takes time to do hard math problems in high school). Andrew Wiles himself took 7 years to prove Fermat’s Last Theorem. Imagine how many year it took to create and prove an entire grade 11 math textbook. It probably took thousands of years to prove everything within that textbook, and yet we are able to condense this into a single textbook. The knowledge of mathematics we gained through time really does take a lot of time, and is simply amazing.
I think this story would interest students at grade 10-12, and I would show this film once they know what Pythagoras Theorem is. Why? Well for example, in a grade 11 class, everyone knows that Pythagoras Theorem is a^2+b^2=c^2 for right triangles. But how exactly can we prove that this is true? How much time would a person have to invest to show that this theorem is true, so that this knowledge could be used for future generations? This video would be a perfect example to answer those questions.
I think that including mathematics history in math classes is a great idea, and it might spark interest for students who like history. I remember in class we looked at a book called “zero”, and it talked about how people back in the day thought that the number zero was non-sense. How cool would that be if I taught that to students? How cool would it be if I said that back in the day, negative numbers and complex numbers was considered absurd to most people? Not only is it cool to teach mathematics history, but it is to also appreciate the people in the past who has gained this knowledge for us.
Arcavi et al: Why Teach Math History
Back in high school, I also wondered why teachers did not incorporate math history into the curriculum. As I did pre-reading of the article, they listed that one of the objections was that “History is not mathematics”. I guess this is true in some sense because they are two completely different subjects. If you want to learn about the history of mathematics, then maybe they should have incorporated that in History class as opposed to Math Class. However, I feel like that would be unfair. In music class, not only do they teach you the instrument you play, but they also teach you the history of famous musicians like Mozart and Chopin. Why does the music curriculum incorporate history, while the math curriculum is not mandatory to?
The one thing that I really agree is when they talked about appreciating mathematics as a cultural-human endeavour. I especially liked that the part where they said that “mathematics is not a rigidly structured system of results, but a continuously evolving human intellectual process, tightly linked to other sciences, culture and society.” If we show the students the history, time, and commitment it takes to come up with the formulas that we have today, then students will start to see the human development of mathematics, rather than just seeing the formulas as some sort of system. I also agree that by implementing math history classes, we can help students develop personal growth and skills (not math development) such as reading, writing, discussing, and analyzing and etc. These personal growth and skills were exactly what famous mathematicians gained during their journey of discovering of math formulas/theories.
After reading this article, I have started to gain some idea of how I could incorporate math history in a classroom. We may not be able to completely talk about the history of the person itself, but at least we can explain the thought process of developing the formula or theory. Maybe we can explain at what time period it was developed. For example, let’s say we are learning about irrational numbers. Where did the idea of irrational numbers came from? Maybe tell them about the time in Ancient Greek. Why do we need irrational numbers? Because there were some lines that actually can’t be measured with a ruler. Why did people think that irrational numbers were outrageous back then? The idea that a number which couldn’t be turned to a fraction was crazy to them because Greeks thought that all numbers could be expressed as a fraction (therefore rational).
Introduction to Math History From Joseph’s Crest of the Peacock
One thing that stopped me was when they mentioned that the concise and meaningful of mathematics is virtually impossible. I thought to myself, “I wonder what the dictionary says about this.” If you were to search for the definition in the dictionary, it says that mathematics is the abstract science of number, quantity, and space. This is interesting because the definition has not say anything about deriving proofs and reaching conclusions, and it doesn’t talk about sets of methods. Not only that, but it doesn’t say that mathematics is like another “language” that is known worldwide. However, everything I have said only explains a part of mathematics, and not mathematics as a whole. So I guess it really is true that it’s virtually impossible to have a definition of mathematics.
Another thing that stopped me was when they mentioned the 4 thousand year old table. I literally stopped, and googled the n^3+n^2 formula and wanted to see how they solved cubic equations in the form x^3+x^2=c. This is something I never even learned before in mathematics! I mean if they teach you how to solve quadratic equations in high school, why don’t they teach you how to solve cubic equations? This could of been very useful in my Calculus classes if I wanted to find roots instead of using newton’s method or intermediate value theorem.
One thing that surprised me was that the Greeks were not the first ones who “discovered” mathematics. Egypt and Mesopotamia actually had developed written mathematical records before the Greeks, but they were dismissed as little importance to history. Hence, Greek become the source as “creators of mathematics” instead.
Another thing that surprised me was Figure 1.4 in the book. I was surprised to see that India, the Arab World, and China also played a role in the development of mathematics. I was actually curious India’s proto-mathematics from bricks and baths, and what type of algorithms they computed in the classical period. In fact, why were they studying algorithms? Were they building something during that time?
History of Babylonian Word Problems
Pure’ vs ‘Applied’ Mathematics, Abstraction, Practicality
I believe that problems do have to be (in some extent) superficially about the “real world” so that most students can apply their mathematics outside of school setting. Making the questions contrived is necessary to advance the knowledge of students to the next level, so that later on the problems are not so contrived and applicable. I also believe that applying mathematics to students will give them motivation that will be useful to them in the future. That’s why I really liked the idea of how Babylonian schools have word problems dealing with agriculture, construction of buildings, and siege ramps. However, we also need ‘pure’ mathematics because that is the core of mathematics. What I really mean is that applied mathematics cannot exist without pure mathematics. If no one proved all the formulas, equations, and theorems, then how could you apply it knowing that these formulas might not be true? It is true in some extent that a lot of ‘pure’ mathematics is too abstract, therefore useless to real life applications. However the more abstract it is, the higher chance it is for it to be applicable to real life applications in the future. What do I mean by that? For example, those who study number theory have learned about prime numbers and modulo. Do raw knowledge of prime numbers and modulo help apply questions in the real world? Not really. Prime numbers don’t have patterns, so why study them? Well, we don’t know why yet, so let’s go further and see what we can find. Later on, we created a system in which we could actually used prime numbers and modulo to encrypt messages. Now that is useful. Later on, we extended this to encrypt personal information on the internet/computer. Hence, the more we study the abstraction of mathematics, the more chance it has to apply it to the real world.
Revised
It seems that the original post was more focused on the argument between “Pure” vs “Applied” Mathematics, but fails to mention practicality, and generality of word problems, which was the core of the reading.
The question I would like to ask is “What is the point of word problems?”. Do we use it for the sake of evaluating a student’s sense of the concept, and understanding the theory? I believe that we use word problems simply because we use it in our minds everyday. For example when I wake up, I always think about the amount of time I need to get to school. How many minutes will I need to brush my teeth, shower, eat breakfast, and bus? What if the bus driver is driving 3/4 of the speed than the usual speed to get to school? How much more time will it take to get to school? Will I be late? Word problems are just unconsciously start popping out of my head. One interesting book that relates to this is called “math curse”, who talks about a kid who is cursed about thinking about math word problems all day.
I suggest you take a look at it when you have the chance!
Yes, one may say that these word problems may be practical, but the Babylonian word problems in this reading don’t seem to be practical. In fact, it seems that they are creating word problems for the sake of solving them. But what is wrong with that exactly? The more of these type of word problems we solve, the better we solve things abstractly. The better we solve things abstractly, the more easier it is to transition to word problems are that more ‘applied’.
Was Pythagoras Theorem Chinese- Visiting a Old Debate
It is definitely important to acknowledge non-European courses of mathematics. By acknowledging them, we are recognizing that there are other cultures (such as Middle Eastern, Chinese) who have tried to prove the same formulas, or theorems. If we are not acknowledging, them we are simply promoting cultural superiority, and encouraging the idea that only mathematicians in the Western World are the only original contributors of mathematics. It actually kind of made me disappointed in myself when the article talked about the gou-gu theorem (aka Pythagoras Theorem). I couldn’t believe that I didn’t know about this theorem even though I was Chinese! Wouldn’t it be cool if I told my students that there were other cultures who discovered Pythagoras Theorem? I actually think that it will make math class even more interesting! By presenting the different types of cultures who discovered the same theorem, students will generally become more interested because it relates to their own culture. Students will then also see that mathematicians can be any type of race and gender, which gets rid of stereotypes.
Naming math theorems and concepts are important since it is a way of acknowledging the person who found the theorem. It was because of Pascal’s Triangle that we know Pascal, and it was because of Pythagoras Theorem that I know Pythagoras. I honestly am thankful for these people who had advance mathematical thinking. However, I think that we should also give multiple names to the theorem, and not just one. For example, Pythagoras Theorem could be called Pythagoras-Gou-gou Theorem. With this, we could acknowledge both of these people. However if there are way too many cultures, maybe we could call it “right-triangle theorem” instead. And then list the many types of mathematicians who proved this theorem.
Reading on Indigenous Mathematics: Mathematics as Medicine
One thing that stopped me was when he mentions that mathematics is an essentially simple (not complex) way of thinking. I would like to disagree with this, since I believe mathematics can get pretty complex to where no one knows how to solve a problem. For example, look at all the millennium problems that haven’t been solved yet. If mathematics is so “simple”, then why do only a few select people like it? In addition, he mentions that mathematics is all about simplifying, clarifying, and analyzing, and breaking down. The mathematics he’s explaining is deductive reasoning. If we are talking about inductive reasoning, then that would be different. In fact, inductive reasoning would be similar to Indigenous thought (because it’s also about building and developing). Maybe I am misunderstanding something here.
Another thing that stopped me was when they said that mathematics is a requirement for indigenous people to succeed in the job market. He said it’s mostly because of the desperate states of education for Aboriginal people for math and science. This was interesting to me because it seems to be quite the opposite in BC. Those who major in mathematics find a hard time getting a job in BC, unless you are in computer science or engineering. Perhaps we have lots of education in a multicultural country, therefore too many people are already in the job market for most races.
The last thing that surprised me was when they said that one function mathematics plays for Indigenous mathematics is a source of power. I have never thought about it as power, but rather as a source of intelligence. But I guess I can see that intelligence is power in a sense, because with intelligence we can further advance technology. However, the power they are talking about is power within ethical tradition, and not power in an individual person. What was super interesting was when they mentioned Ramanujan and how he got his “power”. Generally, one gains mathematical insight from their own mind and personal self. However, Ramanujan says he got his mathematical insight from dreams of his family goddess.
Numbers with Personality by Alice Major
For me, I have never thought of numbers having personalities. I have always worked with numbers, but I never really thought of seeing numbers as “people” or a symbolization of something. However, I do know other people who believe that numbers represent something. For example, Chinese people think that the number 8 is a sign of “good luck” and the number 4 is “bad luck” because 4 in Chinese represents death. I believe Alice also mentioned some other numbers in Chinese and what they represented, which was pretty interesting. I also read a documentary where there was a guy who was blind, and he saw each number as a color. For example, the number 3 was red, and the number 9 was blue. However, Alice Major’s poem about the numbers were more than just symbolizing them. You could actually imagine these numbers as gods with distinct personalities, what they wear, and what scenery they would be in. I could never go to that extent in imagining numbers.
Honestly, I would never introduce this to my secondary math students. I think personality of numbers is more of an “art” than math. Also, everyone has a different perspective and idea of what a personality of a number is, so it’s really subjective. Mathematics is not suppose to be subjective, it’s suppose to be logical.