Monday, September 30, 2019
Saturday, September 28, 2019
Math Art Project Reflection
Initially, when the course outline was given to me, I was a little confused at the prospect of a math art project. I admit that it didn't appeal much to me at first. However, after completing the project and making connections to math concepts applicable in a high school context, it made me appreciate the intercurricular activity a lot more. I never really had an interest in art, but seeing mathematics visualized in an aesthetically-pleasing manner piqued my interest!
As for the project itself, my group ultimately decided on Margaret Kepner's piece from the Bridges 2019 gallery. She provides two different interpretations of the same concept: sequences signified by colour, one on a square grid, and the other on a triangular grid. After the first day discussing with my group members, I made a suggestion that we recreate the piece using LaTeX, as I was striving for the most authentic replica. During my trial-and-error of programming, I came to the conclusion that the triangular grid was the easiest to implement, and so we focused our efforts on that picture. The programming went relatively smoothly, only hitting a few hurdles like colours and triangles overlapping.
As for our own interpretation, my group members came up with the idea of using both lucky numbers and perfect squares. We also agreed on changing Fibbonacci numbers to be represented by white dots in the geometric centre of the triangle, for better clarity. The idea to idea to use perfect squares was a hidden gem. It revealed a pattern alike to the one produced by the triangular numbers in the original. However, there was a very interesting difference: it consisted of only five arms, compared to the seven of the aforementioned.
This activity was great in relevance to the curriculum as it dealt with different sets of numbers. Students learn about these sets but may not directly see the connection between them. Being able to visualize in mathematics is very important, as some students are not able to fully understand using traditional methods.
Looking back at the presentation, the implementation of the in-class activity could have been handled better. Unfortunately, with the grand scale of the project, it was difficult for the class to draw their own with the limit of 96 triangles. With time in mind, this is really the largest we could reasonably have, but the drawback was that the pattern was hard to see for some of the arms, and it did not come into fruition as much as we hoped. All said and done, I am elated at the finished product, and glad me and my group members were able to display sequences in visual patterns to the rest of the class, and hopefully more of the same is in store for my future students.
As for the project itself, my group ultimately decided on Margaret Kepner's piece from the Bridges 2019 gallery. She provides two different interpretations of the same concept: sequences signified by colour, one on a square grid, and the other on a triangular grid. After the first day discussing with my group members, I made a suggestion that we recreate the piece using LaTeX, as I was striving for the most authentic replica. During my trial-and-error of programming, I came to the conclusion that the triangular grid was the easiest to implement, and so we focused our efforts on that picture. The programming went relatively smoothly, only hitting a few hurdles like colours and triangles overlapping.
As for our own interpretation, my group members came up with the idea of using both lucky numbers and perfect squares. We also agreed on changing Fibbonacci numbers to be represented by white dots in the geometric centre of the triangle, for better clarity. The idea to idea to use perfect squares was a hidden gem. It revealed a pattern alike to the one produced by the triangular numbers in the original. However, there was a very interesting difference: it consisted of only five arms, compared to the seven of the aforementioned.
This activity was great in relevance to the curriculum as it dealt with different sets of numbers. Students learn about these sets but may not directly see the connection between them. Being able to visualize in mathematics is very important, as some students are not able to fully understand using traditional methods.
Looking back at the presentation, the implementation of the in-class activity could have been handled better. Unfortunately, with the grand scale of the project, it was difficult for the class to draw their own with the limit of 96 triangles. With time in mind, this is really the largest we could reasonably have, but the drawback was that the pattern was hard to see for some of the arms, and it did not come into fruition as much as we hoped. All said and done, I am elated at the finished product, and glad me and my group members were able to display sequences in visual patterns to the rest of the class, and hopefully more of the same is in store for my future students.
Wordy Puzzle
In this puzzle, the speaker mentions not having any brothers and sisters, so they are an only child. They talk about a man, and mention that he is their father's son. This shows that the speaker is a man. However, if the aforementioned is this man's father's son, then he is talking about himself, since he has no brothers (his father only has one son)! But who is the speaker talking about? Well, that "man's father" is him! That leads me to conclude that "that man" is none other than the speaker's son.
This puzzle is interesting as it uses language unfamiliar to everyday life. It uses three different generations of people: father, man, and son. It makes you translate what "man's father" and "father's son" mean in this context, in relation to the speaker.
Tuesday, September 17, 2019
The Locker Problem
When this problem was first posed, my immediate thought was to use factoring in order to solve it. Since a locker can only be open/closed by every n-th student, only factors of that number of locker will affect the state. For example, students #1, 2, 3, 4, 6, and 12 will all be able to change the 12th locker, and they are also all of the factors of the number 12. Since locker 12 can be changed 6 times (number of independent factors), it follows that
#1 (closes) - #2 (opens) - #3 (closes) - #4 (opens) - #6 (closes) - #12 (opens)
and so its end state will be open. Noticing a pattern, it is apparent that if the locker number has a number of factors that is even, its end state will be open. On the other hand, if the locker number has a number of factors that is odd, its end state will be closed. This is nice and allows one to categorize a locker by finding its factors; however, after testing this for lockers #1-#17, I noticed another pattern which greatly simplifies this process. Most numbers have an even number of factors, since they all come in pairs. There is an exception to this: perfect squares have a pair of factors that are the same. For example, 9 has a pair of factors of 3. Doing this locker problem, we only take into consideration different factors. Therefore, perfect squares have an odd number of factors, and as such a locker number that is a perfect square is closed.
To summarize for the 1000 locker problem, all perfect squares up to this number will have their respective locker number be closed. This includes locker #1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, and 961. All other locker numbers will be opened.
The following link shows my test cases: Drive Spreadsheet
* The letters of the alphabet show the sequence of which student is changing the locker. e.g. column D is the 4th student going through each locker.
The following link shows my test cases: Drive Spreadsheet
* The letters of the alphabet show the sequence of which student is changing the locker. e.g. column D is the 4th student going through each locker.
Monday, September 16, 2019
Letters to Future Me
Positive letter:
To Mr. de Jong,
I just wanted to notify you that I have just landed my first job as a professional engineer! I went through a lot of pain in my coursework over the years, but I was able to make it through, largely due to the way you taught me math back in high school. The way you were able to connect the subject matter to applications in real life really inspired me to pursue an engineering degree, as I could see the relevance in what I was learning. I hope you continue to teach the way you do, so that other people can pursue their passion.
Negative letter:
Dear Mr. de Jong,
I am writing you today to let you know that your class has taught me nothing. I was able to read a textbook and answer problems without your help, and it really left me wondering why we have teachers in the first place. I was able to be accepted into the local university, even though you tried your hardest to give me a low grade, because "I didn't put in the effort". Well, you have to realize that I have many other courses I am taking, and I don't believe math to be of much importance in my future as a humanities major. Your subject is not that special, and it's my job to make you realize that.
My hope is that students who leave my class will come out of it inspired, even if it is a mandatory class that they initially have little interested in. Being able to instill a passion inside of them is the reason that I plan to teach.
One worry I have is working with students who already have it set in their mind that math is difficult and not of practical importance. These attitudes are set as early as elementary school, and being able to break this idea after years of disdain will be a challenge.
To Mr. de Jong,
I just wanted to notify you that I have just landed my first job as a professional engineer! I went through a lot of pain in my coursework over the years, but I was able to make it through, largely due to the way you taught me math back in high school. The way you were able to connect the subject matter to applications in real life really inspired me to pursue an engineering degree, as I could see the relevance in what I was learning. I hope you continue to teach the way you do, so that other people can pursue their passion.
Negative letter:
Dear Mr. de Jong,
I am writing you today to let you know that your class has taught me nothing. I was able to read a textbook and answer problems without your help, and it really left me wondering why we have teachers in the first place. I was able to be accepted into the local university, even though you tried your hardest to give me a low grade, because "I didn't put in the effort". Well, you have to realize that I have many other courses I am taking, and I don't believe math to be of much importance in my future as a humanities major. Your subject is not that special, and it's my job to make you realize that.
My hope is that students who leave my class will come out of it inspired, even if it is a mandatory class that they initially have little interested in. Being able to instill a passion inside of them is the reason that I plan to teach.
One worry I have is working with students who already have it set in their mind that math is difficult and not of practical importance. These attitudes are set as early as elementary school, and being able to break this idea after years of disdain will be a challenge.
Mathematics and Me
A positive experience during my school years learning mathematics would be how mathematics just seemed to click for me, whereas other subjects did not. The simplistic beauty of logic inherent in math really enticed me to learn more of this subject material, whereas the other subject seemed to be rote memorization. This example is more related to the structure of mathematics rather than a personal experience, but I feel it nevertheless impacted my future decisions. A more personal experience would be when teachers would relate the topic at hand to real-world phenomena, so that I could connect to the idea of it being relevant to my understanding of the world, hence increasing my interest. In addition, integrating technology into mathematics is a big boon, considering the popularity of if today.
The reason I want to become a mathematics teacher is to display this beauty to the youth in Canadian society, so that they too can appreciate the natural allure of the world. As a more personal experience when I was tutoring, being able to guide a student through a problem and them finally understanding the core concept after many efforts is extremely rewarding as an educator, as you feel like you really are making a difference in their life.
The reason I want to become a mathematics teacher is to display this beauty to the youth in Canadian society, so that they too can appreciate the natural allure of the world. As a more personal experience when I was tutoring, being able to guide a student through a problem and them finally understanding the core concept after many efforts is extremely rewarding as an educator, as you feel like you really are making a difference in their life.
Sunday, September 15, 2019
Reflection: The Role of Representation(s) in Developing Mathematical Understanding
Throughout reading this article, there is an overwhelming amount of support for the use of representation(s) in aiding a student through their journey in understanding mathematics. The author makes many arguments within the paper, but the strongest claim that representation is an important asset in math is with the experiment conducted by Tchoshanov. The findings of this experiment was that the group of students taught using different representations scored significantly higher than those taught using a pure approach in either an analytic or visual approach. The students in the representational group were able to switch on the fly between different modes, depending on which was relevant to the question on hand, which ultimately lead them to have a higher understanding of the material.
As mentioned above, the kinds of mathematical representations included in this article are analytic, visual, and in a more broader sense, internal and external. In addition, there are three levels of engagement with representation, namely enactive, iconic, and symbolic. Some examples of representations excluded from this article include videos and simulations to name a few. A coding simulation of particular algorithms to find the derivative of a graph or the area under the curve in calculus could be a huge boon in helping students develop understanding in said topics. In high school calculus classes, at least from my own experience, calculating derivatives and integrals was relatively simple as we only used simple functions with a well-defined rule. Riemann sums is a topic which eludes most students, and showing a simulation of how they help in defining the integral could prove useful. Students could then, in theory, apply this knowledge to more advanced functions.
As mentioned above, the kinds of mathematical representations included in this article are analytic, visual, and in a more broader sense, internal and external. In addition, there are three levels of engagement with representation, namely enactive, iconic, and symbolic. Some examples of representations excluded from this article include videos and simulations to name a few. A coding simulation of particular algorithms to find the derivative of a graph or the area under the curve in calculus could be a huge boon in helping students develop understanding in said topics. In high school calculus classes, at least from my own experience, calculating derivatives and integrals was relatively simple as we only used simple functions with a well-defined rule. Riemann sums is a topic which eludes most students, and showing a simulation of how they help in defining the integral could prove useful. Students could then, in theory, apply this knowledge to more advanced functions.
Tuesday, September 10, 2019
Reflection: Skemp on two approaches to teaching and learning mathematics
Reading through the article "Relational Understanding and Instrumental Understanding" by Richard R. Skemp, a number of ideas made me pause and think deeper into their significance.
There are two meanings to the word 'understanding': relational understanding and instrumental understanding.
The distinction between the two is an important one, as being aware of why you are doing something gives you introspection into your own learning. From my personal experience, most of my mathematics education dealt with instrumental understanding, where I would master an algorithm for a particular topic in order to complete problems. I did not understand why it worked, but it did, which only gave me a little satisfaction, as I remained ignorant to the underlying reasoning.
Different students may be much more adept at a particular way of understanding.
This ties in nicely with the different learning styles available, and which one works best for a particular student. Instrumental understanding may work for some people, especially those who just want to know how to arrive at the correct answer; however, for the others who need to rationalize what they are doing, this is insufficient. This makes me question whether I could have a deeper comprehension of mathematics (and other subjects) if my education was delivered in a different manner, as I frequently caught myself asking 'why' we learned a topic a certain way.
Relational understanding can be interdisciplinary.
This point is quite important, especially with the emphasis in the curriculum with how the different core subjects all relate to one another. Understanding mathematics at a fundamental level can aid someone in a seemingly unrelated topic (such as proportionality). This is directly contradictory to what I believed in growing up, as I held the belief that core subjects have nothing to do with each other, and a different set of skills were required to master them separately.
Skemp's issue of understanding
Delving into the main theme of the article, it is apparent that while the two different types of understanding exist, Skemp clearly favours relational over instrumental understanding. In my opinion, I would tend to agree with him, as being a teacher candidate you should have a deep knowledge about your subject area. One of the worst things you can do is have a student question why you are doing something a certain way, and just replying with "well, that's just the way it works". The student will be left with no sense of accomplishment, as they just feel like robots programmed to answer problems (which is where the popular dislike for mathematics may stem from).
However, I also agree with Skemp's decision to highlight the positives that instrumental understanding brings. Not everyone learns the same way, and I think it is important to tailor your teaching to a diverse range of learners. Ideally, when introducing a new topic you would do so with a relational understanding to be your foundation. Only after this, you can implement instrumental understanding. Doing so would supplement a student's learning, showing them different ways at arriving at the same answer, in a more timely manner as well.
There are two meanings to the word 'understanding': relational understanding and instrumental understanding.
The distinction between the two is an important one, as being aware of why you are doing something gives you introspection into your own learning. From my personal experience, most of my mathematics education dealt with instrumental understanding, where I would master an algorithm for a particular topic in order to complete problems. I did not understand why it worked, but it did, which only gave me a little satisfaction, as I remained ignorant to the underlying reasoning.
Different students may be much more adept at a particular way of understanding.
This ties in nicely with the different learning styles available, and which one works best for a particular student. Instrumental understanding may work for some people, especially those who just want to know how to arrive at the correct answer; however, for the others who need to rationalize what they are doing, this is insufficient. This makes me question whether I could have a deeper comprehension of mathematics (and other subjects) if my education was delivered in a different manner, as I frequently caught myself asking 'why' we learned a topic a certain way.
Relational understanding can be interdisciplinary.
This point is quite important, especially with the emphasis in the curriculum with how the different core subjects all relate to one another. Understanding mathematics at a fundamental level can aid someone in a seemingly unrelated topic (such as proportionality). This is directly contradictory to what I believed in growing up, as I held the belief that core subjects have nothing to do with each other, and a different set of skills were required to master them separately.
Skemp's issue of understanding
Delving into the main theme of the article, it is apparent that while the two different types of understanding exist, Skemp clearly favours relational over instrumental understanding. In my opinion, I would tend to agree with him, as being a teacher candidate you should have a deep knowledge about your subject area. One of the worst things you can do is have a student question why you are doing something a certain way, and just replying with "well, that's just the way it works". The student will be left with no sense of accomplishment, as they just feel like robots programmed to answer problems (which is where the popular dislike for mathematics may stem from).
However, I also agree with Skemp's decision to highlight the positives that instrumental understanding brings. Not everyone learns the same way, and I think it is important to tailor your teaching to a diverse range of learners. Ideally, when introducing a new topic you would do so with a relational understanding to be your foundation. Only after this, you can implement instrumental understanding. Doing so would supplement a student's learning, showing them different ways at arriving at the same answer, in a more timely manner as well.
Wednesday, September 4, 2019
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