Mathematical understanding and multiple representations

 

What convinces (or doesn't convince) you in the authors' argument?

I was convinced that mathematics can have internal and external representations. Internal representation is more abstract whereas external representations can be expressed through different means. Mathematic as a subject that host huge amount of abstract ideas, being able to represent or “externalizing an internal abstraction” is crucial. The author mentioned that the nature of an external representation can influence the nature of the internal one, and vice versa. I think this is true. For example, if the internal concept I have is “set”, then my external representation of a set can be a basket. We can extend the idea of a set to study open and close sets. Since a set can be open or close, a good representation would be a basket with lid. But what about a set that is both open and close? The external representation of using a basket can no longer demonstrate. We would need to change the way to represent. Hence the nature of external representation changes due to the evolving internal concept. 

·       What kinds of mathematical representations are included and excluded in this article? Can you think of an example of a mathematical representation of a particular math concept (from secondary or elementary school curricula) that is not included, but that might be helpful for students in developing understanding? Describe briefly how you might teach using this representation.

In this article, author presented mostly visual representations for mathematical concepts, as well as using patterns as representation (the Fibonacci sequence). One example of a mathematical representation of a math concept can be found in representing polynomials using algebra tiles. The following chart shows the tiles and what they represent.

We can represent a given polynomial expression. We need to determine which tiles we need and lay or draw them out side by side. For example, if we want to express 2x^2 – 3x + 3, we need 2 positive x^2 - tiles, 3 negative x-tiles, and 3 positive unit tiles.

My most and least favorite teacher and letters 10 years from now

My most favorite math teacher

      My most favorite math teacher is my undergraduate senior project supervisor. She gave me a lot of freedom in choosing my topic and deciding the direction of my research. I chose game theory to do research on and it was not her specially (her specialty is in graph theory). She was happy to accept my proposal and told me that I needed to be the leader of learning in this topic and she will try her best to guide me. She was honest about her lack of knowledge in game theory but she was willing to accommodate. I took many classes with her including discrete mathematics, vector calculus and graph theory. She is my favorite math teacher because she always treat students with respect and equality. She valued our opinion on how we wanted to be taught. Most importantly, she was willing to compromise or to change so that students are better assisted. 

My least favorite math teacher

My least favorite math teacher is a very nice lady in high school. She was teaching grade 12 mathematics and I was in her class. In terms of teaching, she seemed just as competent as any other math teachers. Nothing too outstanding but also nothing too unacceptable. During that semester, I happened to be hospitalized for two months. I knew it would be hard for me to catch up a class. I decided to maybe take the grade 12 math some other time. However, I didn’t want to waste the time for that block, I asked that teacher in private about the permission for being an audit student. She rejected me and said she has had enough accommodating students from her class already and she is not willing to do more accommodations. I argued that I will not do any homework or test for her to mark, I will only sit and listen the lectures. She didn’t gave me a clear reason on why having me would be a burden to her. I didn’t have a clue on why she would “compromise” to have me sitting in her class.    

      Letters from my future students, ten years from now.  

Dear Ms. Li,

My name is G, I was your grade 12 math student 10 years ago. I am writing to first thank you for leading my way into mathematics. I wasn’t a really talented math student back then and I was not confident in learning math at all. It was you who encouraged me continuously and provide extra help whenever I needed. Grade 12 year was quite crucial and I was under a lot of stress. I even thought of dropping calculus 12 at one time. However, you reassured me and taught me that quitting cannot always solve problems. I always remembered what you said up to today that I should always be proactive rather than passive. I graduated and got acceptance from both SFU and UBC’s math departments. I chose to study the dual degree program in math and education. And believe it or not Ms. Li, I am now a math teacher like you! I hope I can be someone like you to empower many someone like me 10 years ago. Once again, thank you.  

Dear Ms. Li,

My name is M, I was your grade 8 math student 10 years ago. I am now a student from SFU studying psychology. In grade 8, I remember hating your class. For most of the time, I didn’t fully understand your lectures but I was afraid of looking stupid in front of my peers. Hence I didn’t reach out for help. I was very behind on the course material and I can barely pass my tests. Even though you asked me a couple times to see if I was okay, I was too shy to admit that I had little clue on what was going on. I did pass your grade 8 math class, but there was definitely a huge gap in terms of mastering the knowledge. For the rest 4 years in high school, I had different teachers for mathematics, but the same situation repeated. I was behind more and more each year. I failed grade 11 math at the end and stopped taking math courses. Because of this, for university, I can only apply for art and humanity major programs. And here I am, at psychology. When I though I will escape from my nightmare (math), I realized that even in psychology, math is closely involved. We rely hugely on statistical analysis and as well as inferential statistics. Many of those courses I had to take them 2 times just to pass. In retrospect, I would appreciate if you told me about the importance of mathematics in grade 8. Right now, I have to continue to grad school since bachelors in psychology won’t get me into any related jobs. My friends from high school have already been working as computer engineers, nurses, teachers and etc. I wish I reached out for help when I needed. And Ms. Li, don’t trust student when they say they are okay. For most of the time, they are not.

Discussion on Sept 21

To add on to Sarah's group, I totally agree with them on the first point they made. At the beginning of learning math, it is not practical to start the students with relational learning. Instrumental learning at the beginning serves as foundation bricks helping towards building the structure of math where relational learning starts.

Just to add bit to Jacob's. I think for some respective area in math, usually in higher level, some talents may be counted as advantage. For example in areas of abstract algebra or algebraic geometry, students needs to be extra creative in coming up with ways to prove. However, for the material taught in high school level math, all students should reach understanding with appropriate amount of practice. Student shouldn't use the excuse of having "math anxiety" to avoid even putting efforts in studying.

Just to add on Alexa's group, I think student who graduate high school is unlikely to use the knowledge they have learned in high school mathematics. However, learning math is a way to sharpen student's thinking skills. Student should be able to think independently and logically. I think this is one of the goals in mathematical education. 

The 1000 Locker Problem

 

A school has 1000 students and 1000 lockers. On the first day of school all lockers are open.
Student #1 closes all lockers.
Student #2 opens each second locker.
Student #3 changes the state of each third locker
. . .
And so on until all 1000 students have had their turn.

After all 1000 lockers are done, which lockers are open? Which lockers are close? Why?

Discussion and Reflection on Skemp's Article

 

Education is for every global citizen. So are the discussions on the education. (1) However the jargon heavy beginning part of the article and the use of music lesson example as analogy to ‘relational mathematics’ vs ‘instrumental mathematics’ could make its accessibility limited. (2) Author uses two definitions of ‘football’, namely soccer and rugby, as a supporting example to says it matters when we have two different concepts of understanding. Football example can be viewed as a definition problem in which having mutually acceptable and consistent definition would avoid such problems. (3) The three examples demonstrate how a teacher would teach area of a rectangle, multiplication of two rational numbers and circumference of a circle. They may be a problem of insufficient information. It may be because the term ‘instrumental understanding’ is not a well-accepted and consistent definition yet. These made me stop and think and the next paragraph is a generalization out of (3).

The issue the author raised looks like a part (or parts) of a larger problem. Mathematics is a growing body of knowledge and people contribute their parts in this construction process without a detailed drawing of the entire mathematics building. The builders get more mature as the building gets better and humans gather more experiences. The learners of mathematics also more or less follow this road from behind the builders. In each topic of interest, it is important to declare the set of objects we work with and the consistent definitions along with validity, conditions and logical reasoning. However, due to different maturity levels among learners, teachers would learn to know where/how to start. For example, a definition of the area of a bounded region that is part of a Euclidean two dimensional space is a measure of its extent that we humans want in number. One agreed definition is a multiple of a standard square-unit. This is how I understand from learning, not yet what a pure mathematician would say. Whether we tell such story to a grade 5 student is debatable. According to the mathematical maturity of the students we may start with:

Length and width must be measured in the same unit and are non-negative numbers. This formula is valid only for the rectangular regions laid on a flat surface. Some of its applications are measure of a land plot for pricing, measure of floor area for estimating materials to buy. This can be used as definition for beginners. From that starting point, students can continue to learn areas of different shapes in flat plane using logical reasoning. After having familiar with areas on flat surface, they could generalize the concept, again using logical reasoning, to cover areas of cylinders, spheres and so on. Students should be offered a safe sufficient amount of information depending on their maturity.

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