Reflecting Upon My Reflections

In the first couple weeks of the course, we talked about learning mathematics and what we can do as teachers to enhance our students' understanding of mathematical concepts. The articles read helped me to realize the best approaches to use and to look at my own teaching philosophy critically. We were also periodically given math puzzles to work on and post our own solutions to. This was one of my favourite aspects of the course, because I feel like this is a great way to gather interest in mathematics, which has traditionally been seen as a 'boring' subject, and I will definitely consider implementing this in my own classroom. In addition, seeing all the different types of approaches different people use helps to highlight that there are many ways to solve a problem, not just one. 

The first major project was the math art project. After having completed it, I came out of it with a greater appreciation for math and its intrinsic artistic beauty. It was also a great way to work collaboratively with other teacher candidates while working towards one common goal. It reminded my of myself in Grade 9 Math Honours, where we were tasked with creating a picture on our graphing calculators using common functions. If there was a project aspect of a math course I am teaching in the future, I would like to do something similar.

The next theme was looking into the curriculum and had us do microteaching lessons. Although I found the amount of time devoted to looking at the curriculum to be lacking, I did like the microteaching lessons. The one thing that stood out to me was the timing associated with teaching, as I believe it is one of the most important things to consider when planning a lesson.

We as a class also participated in two math fairs: one we visited, and the other visited us. I thought that these were interesting as it gave me a lens I could look through and see how different students were able to mesh their 'creative' side with their 'mathematical' side. Also, the passion each student had while presenting was great to see!

In the end, I am happy with the course and the activities I participated in. However, since I have two teachables, I was only able to take one math curriculum course, and I wish I could take more! 

Off the Grid

Overall, I agreed with the theme of the paper. Why are so many places separated by straight lines? Although this may seem to be not as important at first, if you look deeper into it the nature of this is problematic. Consider Africa for example, where straight lines or other unnatural boundaries were created by Europeans during colonization. This has ultimately lead to violent ethnic conflict in certain regions, as feuds and relationships between different peoples was not taken into consideration. Or on a lighter note, the Canada-U.S. border is infamously a straight line for the most part along the 49th parallel. This has lead to weird citizenship situations, notably Point Roberts, which is connected to Vancouver Metro, but is part of the United States since it technically is below the 49th parallel. Since the citizens of this city are American, they have to attend school in Washington state, which takes at least 40 minutes!

A point in the article that struck my interest was when the author was talking about the Principle of Relativity and related it to grids in the style of Riemannian geometry. Arguing that there is no right geometry to use, and as such we should broaden our horizons when it comes to implementing different geometries in grids was an interesting statement. They say that "variety is the spice of life", and I feel like giving grids the ability to be designed in mathematical patterns would be fascinating.

When talking about incorporating Indigenous content into classes, I always wonder "how", especially in a mathematics course. However, this article sheds light Indigenous traditions and their relationship with geometry. Being able to use this in a class could help both Indigenous and non-Indigenous students. Indigenous students would be able to learn in a way that they are accustomed to, whereas non-Indigenous students would be able to explore new ideas of how to appreciate and solve problems in math, and being able to solve problems in a multitude of ways helps the student understand the content better.

Reflection: Math Fair

I thought that this math fair was quite interesting. I thought that representing mathematics in a way different than that taught traditionally was a breath of fresh air, especially since many people claim that math is boring. The students also seemed to be enthusiastic about presenting their hard work, as it's apparent they worked very hard on their project, and were eager to display their work they invested so much time and effort into. One thing that caught my eye was the project about changing algebraic questions into work problems. When I think of common student difficulties, what comes to my mind is word problems. But these students found it useful to first translate the question into a word problem, and then solve the question, which was an interesting insight into how different students may approach the same problem.

Drinking Party Puzzle

Starting with 2 bottles, a rat can choose to drink from either the first or the second bottle. If it drinks from the first bottle and dies, then that one is poisoned. If it drinks from the first bottle and lives, then the other one is poisoned.

Anything above 2 bottles will require additional rats. The greatest amount of bottles tested with 2 rats is 4 bottles. Let

• rat #1 tests bottles #1,2
• rat #2 tests bottles #1,3

then

• bottle #1 is poisoned if rats #1,2 die
• bottle #2 is poisoned if rat #1 dies
• bottle #3 is poisoned if rat #2 dies
• bottle #4 is poisoned if NO rats die

For three rats, I found the maximum number of bottles tested to be 8. Let

• rat #1 tests bottles #1,2,3,4
• rat #2 tests bottles #1,3,5,7 
• rat #3 tests bottles #1,2,5,6

so that all rats share one bottle (#1), all rats don't drink from one bottle (#8), all rats drink from one bottle which no other rat has drank from (#4,6,7), and two rats share the same bottle (#2,3,5). This covers all the bottles. Then

• bottle #1 is poisoned if rats #1,2,3 die
• bottle #2 is poisoned if only rats #1,3 die
• bottle #3 is poisoned if only rats #1,2 die
• bottle #4 is poisoned if only rat #1 dies
• bottle #5 is poisoned if only rats #2,3 die
• bottle #6 is poisoned if only rat #3 dies
• bottle #7 is poisoned if only rat #2 dies
• bottle #8 is poisoned if NO rats die

A pattern is starting to emerge. I notice that rat #1 just needs to consume half of the bottles, #1-n/2. Rat #2 needs to drink from every other bottle. Rat #3 needs to drink from every 2 bottles. With 1 rat, it can test 2^1 = 2 bottles. With 2 rats, they can test 2^2 = 4 bottles. With 3 rats, they can test 2^3 = 8 bottles. Therefore, I claim that with 10 rats, they can test 2^10 = 1024 bottles, sufficient for the 1000 bottles of the problem. Continuing from above, rat #4 would drink every 3 bottles, rat #5 from every 4 bottles, etc.

*This works fine for 10 rats, but breaks down for 9 rats.

After thinking more about this, I can show that the number of bottles follows 2^n, where n is the number of rats. Let R = {the set of rats}. Then the power set of R, P(R), lists each bottle and which rats drank out of them. For example, for n = 3, we have R = {1,2,3}, and so

P(R) = {{},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

From this, we see that each subset matches up exactly with the hand-written solution above. We can then use a theorem that states: "a set containing n distinct objects has 2^n subsets". Therefore, the power set of a set containing n distinct objects has 2^n elements. And so for 10 rats, n = 10, and the set of bottles has 2^n = 2^10 = 1024 elements, which means that 1024 bottles can be tested.

Unit Plan

Unit Plan

*updated December 1st

Thinking About Math Textbooks

When I read through the article about first and second person pronouns, I was surprised to read that first person pronouns were not in either of the textbooks that were used for the purposes of this article. When I think back to my personal experiences with mathematics textbooks, most if not all examples of person pronouns involved the word 'we', which would fall under first person, NOT second pronoun. Perhaps it is my recency bias playing a role, and undergraduate math textbooks use these first person pronouns, whereas secondary textbooks still exclusively use second person pronouns. If so, then I must ask, why? The article states that second person pronouns helps to connect the speaker to the reader directly; does this only matter for adolescents, and not for young adults?

I understand that today there is a push to not use textbooks, and some teachers are slowly becoming less dependent on them. I've also talked to some teachers about this topic, and it's apparent to me that it is quite the controversial one. However, I still think that textbooks serve a purpose and shouldn't be rid of entirely. Of course, textbooks are expensive and a new version with minimal changes seems to come out every few years. But they also contain a wealth of information in an organized structure, with many additional resources embedded in them. I will say, I think teachers should opt to make their own examples and not 'teach to the textbook', as it allows for them to adapt to their unique class and their needs, compared to the general examples and rigid format that a textbook supplies.

The Scales Problem

We have four different weights to choose from. These will be labelled as a, b, c, d, with each letter be written in ascending order of the weight. Since the vendor sells herbs ranging in weight from 1 g to 40 g, the smallest weight must be 1 g (i.e. a = 1 g). In addition, the sum of the weights must add up to 40 g, such that

a + b + c + d = 40 g

Now, in order to measure 2 g, we could say that the second lightest weight is 2 g. However, another way of doing this is by letting a mass of 2 g be achieved via a weight on both pans. Therefore, a subtraction is needed, and so we have b - a = 2 g. Since a = 1 g, it follows that b = 3 g. With a and b solved for now, I can measure weights of 1 g, 2 g, 3 g, and 4 g. But how can I measure 5 g? Well, the third weight, c, will allow me to. If 5 g is the smallest weight I have yet to obtain, then I must subtract both a and b from c in order to achieve it. So c - a - b = 5, or c = 5 + a + b = 5 + 1 + 3 = 9. This just leaves d now, which can be solved for from the above equation: d = 40 - 1 - 3 - 9 = 27. Thus, the value of the four weights are 1 g, 3 g, 9 g, and 27 g. Looking at these values, these numbers didn't come from nowhere, as it is quite apparent that they follow a distinct pattern: 3^(n-1), where n = weight number. A quick check shows that these four weights account for all the different weights from 1 g to 40 g.

I don't believe there are several correct solutions, as no other way will allow you to account for each weight distinctly, but I could be wrong as I did not delve too deep into proving it.

This puzzle could be extended by looking at different maximum weights (1 g, 4 g, 13 g, 121 g, etc.) and noticing the pattern for which weights you need, and then trying to deduce the formula that tells you the specific weights you need. This could tie in quite nicely with a unit on sequences and series.

Three Curricula All Schools Teach

The first aspect of this article that stood out to me was that about competitiveness in the education system. When I was in high school, me and my friends would always compare our test scores with each other. This always felt like playing a sport, in which you have a drive to win, although it was more of a friendly game. I think that for some students competitiveness may be a good thing, to give them that little extra push to achieve marks. However, the motive behind this may be a little short-sighted. If kids are trying to get high marks due to wanting to one-up their friends, are they really studying for the right reason? Are they really learning? Competitiveness also has a dark side to it. Many students may feel inadequate if they do not perform well relative to their classmates, which could mess with their psychological well-being. As an extreme example, Japanese students who do not achieve high enough marks and as such are subsequently rejected from the top universities, may commit suicide due to the shame they feel, for not living up to expectations set by them or by others.

The overall theme of this paper was to look at what secondary schools (in the U.S.) teach, and what they do not teach, and the overall structure of the curriculum.  It never quite sat well with me that in B.C. we must complete an English 12 in order to graduate, yet no other grade 12 course was needed. Even further, it was the only provincial exam in grade 12, and was worth 40% of your final mark (at least it was when I took it in 2013). This caused me great stress as English was my worst subject, yet I had to take it, and universities placed a great emphasis on having a good mark in this particular subject, no matter what program you apply to. "Why were arts students not scrutinized by their math marks?" was my thinking. Was success in poetry a good indicator of a student's aptitude in post-secondary education?

It seems to be that the core subjects consist of language arts, social studies, mathematics, and science. Any other subjects are more fringe and are not guaranteed to be offered at every school. Yet it appears odd to me that astronomy is not taught as a science 11 or 12 course in the B.C. system, despite it being the study about the entire Universe. Instead, a small topic is inserted into the Earth Science 11 curriculum, but I don't think it does the subject justice. There is the option in the new curriculum to make the course yourself (as a teacher) under Specialized Science 12, but the issue of it not being offered at many schools still stands. I do admit that I am obviously biased regarding this topic, but to me it always seemed like the astronomy portion of general science courses is the one that teachers often gloss over and pay little attention to, and I just wonder, why? Perhaps the reason as to why I decided to major in astronomy was my yearning for a proper learning of what the subject really has to offer, since I was not given the opportunity in my secondary education.

Reflection: Curricular Microteaching

This microteaching lesson was the first one in which we looked at the BC curriculum, and as such it felt very applicable to a real secondary school classroom. My group decided on tackling the transformation portion of Pre-calculus 12. In particular, we looked at the effect of transforming f(x) to f(ax). We felt like this aspect of transformations would be the best as students historically have trouble with this, as many believe that increasing the coefficient a would widen the graph by that same factor.

We initially looked at the quadratic function y = x^2 and the sinusoidal function y = sin(x). We later came to a group decision to focus our efforts solely on the sinusoidal function, at the advice given by Susan, on the basis of time constraints. We felt that the sinusoidal function would be more illuminating in demonstrating the effect of changing a, as the quadratic function could be argued that you are stretching/compressing vertically (e.g. y = (2x)^2 = 4x^2, so you could say that the graph was stretched by a factor of 4 vertically, rather than compressed by a factor of 2 horizontally.)

We also wanted to represent this transformation using different methods. I made an interactive plot that could switch between sinusoidal functions with different values of a, to display the transformation graphically. I asked students the similarities and differences as a means to engage them and get them thinking about how this particular transformation changes different aspects of its base function. I felt like I did a decent job at having them be active participants in the lesson.

Some things I would work on for future lessons would be the incorporation of applications to make it more apparent to students as to why they should be interested in learning this material. In addition, general whiteboard layout and deciding on having the projector screen up or down with respect to students and their potential issues seeing the board are also things I will look out for.

Curricular Microteaching Lesson Plan

Lesson Plan

Interactive Plots

A Geometric Puzzle

With the numbers going from 1 to 30, there will be 30 distinct numbers, equally spaced around the circumference of the circle. In order for a number to be diametrically opposite to 7, it must be in-line with the diameter of the circle with one end on 7; this occurs when that number is half-way around the circle relative to 7. Since there are 30 spaces, half-way would be 15 spaces, and so we add 15 to 7:

7 + 15 = 22

which tells us that 22 is the number. I also decided to prove this geometrically/computationally. Using the tikz package in LaTeX, I was able to construct this problem with a grid line to further emphasize the opposite number to 7, which is of course 22.

Code:
    \begin{tikzpicture}
        \def \n {30}
        \def \radius {5}
        \draw circle(\radius)
              foreach \s in {1,...,\n}{
                  (-360/\n*\s+84:-\radius)
                  circle(.4pt) circle(.8pt) circle(1.2pt)
                  node[anchor=-360/\n*\s+84)]{$\s$}
              };
        \draw[help lines] (-5,-5) grid (5,5);
    \end{tikzpicture}

You could extend this puzzle for sure, either possible or impossible. To make it impossible, we can make the numbers run from 1 to an odd number, in which there are no opposites. I think there are some scenarios in which impossible puzzles may be appropriate; the act of finding out a puzzle is impossible could be viewed as solving it in a way, as it shows that the student understands the problem well enough to realize there is no solution.

A geometric puzzle can usually be solved using diagrams, whereas logical puzzles require more number theory.

Battleground Schools

At the beginning of the article, the contrast between the two polarizing views of mathematics really caught my eye. Politics is an aspect of life that really envelops a lot of society, and many people feel passionate about their viewpoint over another. Here it is highlighted as such, even in more (or so I would've assumed) apolitical topics such as mathematics. It just goes to show that people will always have different viewpoints, and debates will always be had when it comes to education and the most effective method of instruction.

The second part that resonated with me was when the article talked about math being for 'nerds'. Throughout my post-secondary education, when talking to people outside of a school context, the topic of my studies would come up. I would tell them the courses I was enrolled in, and they would always reply to me that I must be really 'smart' for taking all of these math classes. I always thought, why is it that math out of all the core subjects is seen as for 'intellectuals'? Does this say something about the way our society views mathematics? I can't really say for sure if I am as smart as people believe I am, I just happened to click well with mathematics and less so for the other core subjects taught in the public school system.

On another note, I found it sad, yet humorous, that the catalyst as to why the US decided to overhaul its mathematics education system is due to the "Red Scare" of the Soviets making advancements in scientific technology. Never mind worrying about the education of the youth for the students' benefit; the government was just eyeing the potential benefit these young kids could one day provide to them, which ties into the military obsession that America still has to this day (serve your country!).

The Dishes Problem

The first thing I did when I saw this problem was set up an algebraic equation, where n is the number of guests. I looked for the sum of the dishes, with each term representing a different dish (e.g. n/2 is the number of dishes of rice, since every 2nd guest gets one):

n/2 + n/3 + n/4 = 65

Multiply both sides by 12:

6n + 4n + 3n = 780

Gather like terms:

13n = 780

Divide both sides by 13:

n = 60

And so the answer I got was there were 60 guests in total.

I think it is important to offer puzzles from different cultures, so that students are able to understand mathematics from a different perspective. It is very helpful for students from that particular culture as most of the problems they deal with are from the Western culture, and may not reflect their experiences in life. Showing them that their stories have a place in the classroom too is imperative to show that they and their background matter. 

I think a story helps with setting up the problem. A lot of students complain that math is too rigid and boring, and if every problem just had a simple algorithm to solve I would tend to agree. However, problems with stories attached help in both visualizing and enticing the reader to solve the problem, as it may give the student more enthusiasm since they are invested in the story, and want to find out the conclusion.

Reflection: Microteaching

The big takeaway from this lesson for me is that 10 minutes is not a lot of time. The most common suggestion I had was to allow for a bit more time for people to write their name. When preparing, I gave 3 minutes for my group to write their names, which was manageable. However, I ended up teaching about 9 people, and as such some students needed help but I was off busy helping someone else. I will give myself the benefit of the doubt considering I was not aware of the amount of people I would be presenting for (assumed to be 3). I made the decision to emphasize Japanese linguistics by getting everyone to repeat some sounds after I explained them, in order for more participation with the students, so that they were not just passively listening. Overall, I am very pleased with how many people enjoyed the lesson and the amount of people that were able to successfully write their name!

Microteaching Lesson Plan

LESSON PLAN: How to write your name in Japanese

Time: 10 min

Objectives: 

• Learn Japanese pronunciation
• Understand Japanese kana and rules with changing sounds
• Be able to write your name in Japanese 

Materials:

• Whiteboard & markers (teacher)
• Pen & paper (students)

Beginning (1 min)

• Greeting and objective in Japanese then English
• Ask anyone if they know what language I am speaking/writing
• Mention Japanese has 3 writing systems; we use katakana for foreign words
• Tell students to consider how their name is pronounced, not spelled

Middle (6 min)

• Go over the katakana chart
• Explain how to pronounce each sound (emphasize 'f' & 'r')
• Talk about voiced sounds ('g', 'z', 'd', 'b', 'p')
• Mention blended sounds (*ya, *yu, *yo) giving examples
• Explain double consonants (tt* = ッ*) and vowel prolongation (horizontal bar ー)
• Indicate what to do with single consonants (* = *u, except for special cases 'n' & 'r')
• Talk about special cases with no Japanese equivalent sound ('v' = ヴ, 'l' = 'r', 'w' = ウ)

Ending (3 min)

• Ask students to get out a pen and a piece of paper
• Have the students write their name using transliteration
• Assist students having issues

Microteaching Topic

My topic will be "Learning how to write your name in Japanese".

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.

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.

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.

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.

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.

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.

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.

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