Re: Mathematical understanding and multiple representations

 I totally agree with the author on how graphs are not just a static end result or an answer, it should be a vehicle for a better understanding and establish a relationship between your answer and a social context. My experience totally encoded with this statement. For me to fundamentally understand and apply mathematical phenomena, it's curial to visualize the question and have the ability of interpret, construct, and operate effectively with both external and internal representations. There is a lot of concept-based knowledge in math that is hard to "show". Our brain just works better when we can realize the pattern so that we can encoding and decoding the information. To have a thorough understanding of a math problem, students should be able to not only "solve" the questions but also "make-up" similar questions as well. 

The author mentioned multiple representations include graphs, symbols, diagrams, physical and virtual manipulatives, etc. One presentation I think would be helpful to students to develop their understanding of the concept of distance is sound. Humans have the ability to distinguish the loudness and frequency of the sound depending on how fat they are from the sound source. Doing an in-class activity of such a presentation will help students to notice the relationship between distance and math in our daily life by exploring the representation of sound and distance. 

Reponse to group discussion on Skemp's article

 Re: Sarah's Group 

I like your second point, using math to train your logical thinking skills. I often find that I apply my mathematical thinking skills to my other subjects as well.

Re: Tyler's Group

I read a lot of different opinions on how should we apply instrumental learning and relational learning in terms of in which order, do we combine the both, and which one is more important? I like your point that actually the stronger foundation would come through a relational understanding of the topic!Going from concepts to facts would be easier in my opinion since it won't be so dry and boring.

Re: Homework writing to be completed by Wednesday September 23 at 8 pm

 1. 

One of the math teachers who I didn't like was my elementary math teacher. I'm not sure whether it's because of her teaching style or materials. Her teaching style was mainly instrumentally and asking us to memorize formulas without giving us any reasonings behind them which was very challenging for me. Personally, It's easier for me to understand the foundations of problems through relationally learning rather than instrumentally.

On the other hand, my favorite math was my Grade 10-12 teacher. He made the effort to show students of HIS thinking process which made me feel we were solving problems together. Knowing he also had false assumptions and multiple times of trying really made me understood how everything came together and paid attention to the mistakes he made. He also made sure to show us multiple methods to solve the same problem which encouraged us to understand the question from different perspectives. 

2. 

Dear Zoe, 

Thank you for being my math teacher in high school. You really made a difference in my life! I really hated math before having your class in Grade 8, I could tell you cared about us not just as students but as your friends. 

I appreciated your dedication and the effort you put into the class. How you approached math was different from all other math teachers, you made it fun so we wanted to learn; you made it easy to understand by linking mathematical problems to our daily life. You made me notice that math is everywhere in our lives.

Although, I'm not doing anything related to math as my career. You taught me how to take a like to math and things got much easier when I became interested in it. 

Bests, 

Zoe 

Hi Zoe, 

I'm hoping that what I'm about to say won't offend you. I thought it might be helpful in your future teaching career if I bring this up. 

I found your teaching style was rather boring and inflexible. Your teaching plans were pretty much all based on facts which were really challenging for me memorize, I never really understood any reasoning behind them. After taking your class, I was scared to take any other math courses at the university.

I just hoped you would make your course planning more fun and encouraged students to work on some projects rather than just doing calculations. Students want to learn a skill from your class not a pile of formulas without any actual meanings. 

Thanks,

Zoe 

Re: Our first class math puzzle: The locker problem (for Monday September 21)

It was actually my first time seeing this question. First all, I made sure that I understood the questions correctly which is so curial for this kind of question because if you did the first step wrong, you will basically never get it right. As 1000 doors are a huge number, so I took a smaller approach for just10 students and 10 doors. Please see attached pictures for reference. I'm sure there is some kind of pattern there. 

 

After looking at my first table, I noticed that No. 1,4,9 doors are the only doors that are closed, so I thought it would be easier to find the pattern of closed doors than opened doors. Then I asked myself, why 1,4,9? Is there anything special about these numbers? They all have a factor of 2? Nope, not 9. 1+3=4, 4+5=9, doesn't seem like there are any patterns there. 1^2=1, 2^2=4,3^3=9! Ah, there might be something here! In order to prove my prediction of the pattern is correct, I then wanted to make a table for 20 students and 20 doors, if my guess was correct, then the next closed door should be Door No.16! 

 

Yes! Door No.16 is indeed a closed door! There we go, the pattern of closed doors is the squared number starting from 1, but how do we know until which number is closest to 1000? We can simply just take the square root of 1000, and we get approximately 31.6 which indicates that the last squared number should be 31, which gives us 31^2=961. There we have it, the last closed door is Door No.961, and any door between Door 1 and Door 961 can be simply calculated by using 1^2,2^2,3^2,4^2, etc. til 31^2. Please see my draft for more detailed answers. 

Thus, the Closed doors are Door No. 1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,289,324,361,400,441,484,529,579,625,676,729,784,841,900,961

All other doors are open! 

Response to Relational Understanding and Instrumental Understanding by Skemp

The first thing that made me stop and think while reading the article, was the student asked the teacher to explain the area of the rectangle and he just simply answered along the line of “The formula tells you that to get the area of a rectangle, you multiply the length by the breadth.” and the student said, "I see". That makes me stop and think about the way I was taught in school and also the way I learned. The old educational system was designed to be a "grading system", students just want to know "how to do it" instead of "how to do it" and "why to do it".

Most of us were taught instrumentally. We just want to get it right and pass the test. In the short-term, it seems alright. Both the teacher and the student are happy both they are both teaching/learning instrumentally. The problem is that in the long-term, the student learns ever more rules and developing a shallowing understanding of math concepts instead of understanding the deeper meanings/the whys behind it. They then get confused when seeing questions that aren't fitting their rules. This type of learning/teaching is not sustainable, the students will drop math as soon as they pass the exams or graduate from schools.

As an educator in nowadays, we want to encourage students to have their own critical thinking skills, to go a step further and ask the questions of "why", to self-explore when they have questions, and to learn "how to learn" instead of learning "how to do".