Reflection on The Dish Puzzle due on Oct 5th

            I think this is a famous question in Ancient China. I would solve this question by using the number theory method. There is a total of 65 dishes. Every 2, every 3, and every 4 share a dish, I need to find the least common multiple of 2,3 and 4, which is 12. so if I have 12 guests, then they will share 6 of the first dish, 4 of the second dish and 3 of the third dish, so it will sum up to 13 dishes. We have 65 dishes in total, so we need 5 groups of 12 guests. Therefore, it will be 12*5=60 guests.

          In my opinion, it makes a difference by using a combination of games, puzzles, histories, and stories that are from different cultures. First of all, students could learn about other cultures while learning mathematics. Secondly, the given different types of examples will interest students from diverse cultures with varying preferences for food and hobbies. Compare to pure number problems, word problem contains context about history, stories, and cultures. It often helps the students make a connection from the mathematical concept to the real world. It also allows students to utilize their imagination and develop their creativity. Once students enjoy the topic, they most likely would be curious about it and spend more time trying to solve it.

Reflective Journal- Mathematical art project

 

           Assignment #1 reflective Journal

         When I looked at the earrings art, I felt the pattern was fascinating and wanted to learn more about it. The earrings look beautiful and elegant in a unique way, but I would never know a mathematical concept behind it if I don’t stop and pay attention to it. I think this also one of the purposes for owner Emily Connor who designed them. In her reply to Cheryl’s email, she said, “by interpreting mathematics in different forms and arrangements. We can share mathematics with a wide variety of people and learn more about the mathematics behind our art pieces”. I intend to share the intangible, abstract mathematical ideas blending in arts around us with my students. I believe creating real-world representations of mathematics can raise new questions whose answers have tangible meanings. Mathematical art is especially great for students who might be a little bit wary of mathematics.  

             

         I start to check the Sierpinski triangle pattern used for the pair of earrings. The earrings are made of equilateral triangles and squares with the self-alike never-ending fractal patterns. I have calculated the area and perimeter in the first few iterations and generated the perimeter and area formula. It is then interesting that the remaining perimeter approaches infinity while the area approaches zero after infinity iterations. I extended the mathematical concept from 2D-3D to build a Christmas tree by using a backward procedure of the Sierpinski Triangle. It is 2 iterations Sierpinski project using a total of 16 small tetrahedral prisms and a 3D folding star.  

            I plan to do the 3D Christmas Tree project with my students for learning perimeter and area and reviewing the ratio and fraction concepts in Grade 8 and learning surface area and volume in Grade 9. Through this project, I hope my students can visual and solve the area, perimeter, surface, and volume related to real life. For example, how many color papers we need to buy if we want to build 5 feet tree tall for our Christmas party in class and how many small tetrahedral prisms we need to make together). I plan to introduce the limit concept to challenge them to think about what the final remaining perimeter and area will be and extend the Sierpinski triangle idea to fractal concept and encourage them to find the fractal examples to connect mathematics to nature.

             

 PowerPoint: https://docs.google.com/presentation/d/1valZX_GzR-THyo3PPtRmC9tK4EdU5lcM/edit#slide=id.p7

Entrance Slip Sep 27th– Mathematical understanding and multiple representations

          No doubt that internal and external representations are not separable, and representations help with understanding the abstract concepts in mathematical learning also with any other field of learning. The internal mind can be studied by learning the structure of external representations. The internal representation can be beneficial if the benefit of using external representations can offset the cost associated with the externalization process. This article reminds me of when I first interact with Canadian elementary mathematic textbooks regarding using visual representation symbols or concrete materials of base-ten blocks and rods in place value and addition. I felt the representation was unnecessary since I was so used to the way I was taught which didn’t involve so many visual representations other than using the place value of the positioning method in addition concept. I didn’t get the pedagogical theory behind the Canadian textbook taking large space by drawing many blocks to demonstrate the idea of addition and regrouping. I was concerning kids would feel boring and tired by counting or drawing those blocks. After I start tutoring, I start to understand that every student is different in terms of accepting concepts and build relations between internal and external representations. In my opinion, as educators, we don’t have to necessarily agree with every way of representation for teaching, but we need to have them on our teaching toolbox to help with students’ learning.

          

         This article includes many ways of external representations including graphs, pictures, concrete materials, models. I feel one element I want to add is Venn diagrams because  I think it is a useful and easy visual representation for problem-solving in finding possible logical relations between a finite collection of different sets. For example, it can be used for finding the relation between a set of numbers such as whole numbers, natural numbers, integers, rational numbers, irrational numbers, imaginary numbers, and so on.

The Letters from my future students

Hello Ms. Fan,

I am a student of your grade 9 students. It has been 10 years from when I was in your class when you were in a new math teacher in my secondary school. I have graduated from UBC now and working as an Electronics Engineer. The memory of my grade 9 math class is still fresh to me. I still remember a time when you started to teach us in the fall term, all my classmates were so quiet because we were not used to your teaching style cause we were so used to listen to a lecture and solve exercise problems in the textbook style. You asked us to create our own model to measure and do the experiment when you try to teach us the surface area and volume of cube, cylinder, cone, and sphere. We were laughing at ourselves about the imperfect models of cones and sphere but that was our first math class that all of the class were so engaging. Since then, you, as a new teacher started to connect to us in learning. I am thankful that you build my confidence in math and spark my passion for exploring mathematic. This passion leads to my success in my studies in Engineering and my career as an Electronics Engineer.

Best Regards,

Your Student

Hello Ms. Fan,

I am a student of your grade 12 students. It has been 10 years from when you were teaching me to grade 12 math. I was totally lost and disliked your math class about you don't give direct answers to us, but you were trying to let us find answers by ourselves by trying to engage us in understanding the concepts. I felt so frustrated because I was hoping to get high marks for my math 12 then it can help me to get into a good University, so I stopped trying in math. As a grade 12 student, I was so busy with every course. I did not have time to understand the concept, but I only would like to know how to solve the problem and get the quick correct answer. I managed to get into university and major in Fiance and graduated a few years ago. With the limited understanding of the mathematical concept, I am struggling with my CFA exam, case studies involving mathematical modeling. I wish I were more patient 10 years ago and brave enough to communicate with you about my thought, and you are more experience with teaching grade 12 students, then we would have a more effective teaching-learning Journey. I decided to share my experience with you 10 years later since I saw your passion for teaching, and  I hope it helps in your teaching Journey. I believe you are a great teacher.

Sincerely,

Your student

My Maths Learning Journey

I can not remember how many teachers I've had in my life. However, besides many of my great teachers in schools, I think my favorite teacher is the one who taught me maths in my early life and built confidence in me, it is my older sister. I remember she taught me counting and skip counting, backward counting, simple additions, and etc when I was really young. However, this benefited me when I did an interview for enrolling in school. I still remember the shining face of the interviewer who impressed by my fluent backward counting down. This simply gave me confidence about numbers and maths at a very early stage. Another great teacher for my math learning I think is my best friend from grade 5 in elementary school. In grade 5, I was doing well in maths for academic requirements without knowing much about this subject outside of the curriculum.  However, my best friend was so into groups involving math games and Olympic competitions math outside of math class and she got a big reward from it. She gave me the inspiration to explore other interesting problems and patterns in math beyond the classrooms. I wanted to have fun learning math just like my best friend did. This inspiration helped me learning mathematics with a passion.

Think of the least favorite teacher, I don't have a particular least favorite teacher I remember but  I would say I heard one of my students said he had not had a great 6 and 7 math teacher who didn't teach much for the years which causes the student had a big gap in some math concepts when he went to grade 8. I think this related to responsibility which is one of the most important qualities a teacher must-have. Because of the irresponsibility of the teacher, the students may lose interest from this subject or might have never caught up with their learning. 

By thinking of the great teachers and not-so-great teachers, helping me to imagine myself as a new teacher to be a role model and to be responsible, encouraging, and inspiring my students when they have their math learning journey with my own as a teacher together.

 Exit Slip- Response to the group Comment by (Sarah, Jennifer, Marius, and Ivan)

I agree that relational learning and instrumental learning are both needed in teaching. Relational learning helps with understanding the concept and improving critical thinking and problem-solving skills. This reminded me of when we watch the video about multi-dimensional learning that the students were so engaged when multi-dimensional teaching was approached. Teachers can totally shift students' minds.

I also have the same point of view about math anxiety is the same as from home.  Students need to be encouraged by teachers and parents. A fixed mindset needs to be broken before going forward. 

Locker Puzzle

• Student #1 closes each locker.
• Student #2 opens each second locker.
• Student #3 changes the state of each third locker (i.e. opens them if they are closed, or closes them if they are open)
• ...and so on, until all 1,000 students have had their turn.

After all 1,000 lockers are done, which lockers are open? Which are closed? Why?

      

After I read this question, the first mathematical thing is looking for patterns by induction. 

1st  try: I used 6 students, 

I found 2 locked, the position of #1,#4, the rest of them are open.

2nd try: I  used 10 students,

I found 3 locked in this try, the position is #1,#4,#9, the rest are open.

I see some pattern here, the difference the consecutive number is 3 and 5. they are odd and the difference is 2, so I hope the next difference will be 7.

I also found  I only need to draw the diagonal states since under the diagonal the states don't change any more. This reminded me of matrix diagonal, but I could not go further. 

                          #2 student has multiple of 2 positions change its state,

                         #3 student only when the position number is the multiple of this student number 3, then change its state and so on...doesn't help much

             

3rd try: I used 20 students, 

I found 4 locked, position #1,#4,#9,#16, the rest are open. If we order the position number to locked number  n=1,2,3,4... then the differences between the two consecutive position numbers are 3,5,7...which is 2n+1. I predicted that the next locked position will be #25,#49,#64 since we keep adding 2n+1 to the previous number, then we get a series of perfect square number n^2. 

 I have n^2+(2n+1)=(n+1)^2<=1000, so we have (n+1)^2<=1000, then n+1<=sqrt(1000)

so n<=31.6-1, therefore, n=30, so the last locked position will be (n+1)^2=31^2=961

therefore, it has a total of 31 locked and all the locked positions are the perfect square. The last locked is the #961.

Then I come back to the 1st try to find out the reason. I go back to the idea that #2 student has multiple of 2 positions change its state, #3 student only when the position number is the multiple of this student number 3, then change its state and so on...then the factor idea come to my mind. when having even numbers of factors which mean even numbers of changing states, so it will be open, otherwise will remain closed. Only perfect square numbers have odd numbers of factors, here you go! 

To solve this question, I  used the induction method, using mathematical knowledge about factors, perfect squares, multiples, inequality, patterns. 

Hello world! nice to meet everyone

EDCP 342 Relational Understanding&Instruental Understanding-#1  Entrance Slip

This article opened my mind about mathematical education pedagogy. From my experience as a high school math tutor, I always feel there is something missing upon my students' understanding of math concepts when I try to help them, but I have never had a clear definition of what it was. With the concrete example, the road map Skemp gave, my questions were answered. I totally agree that without relational understanding, learning mathematic won't go far.

I often have students mess up with 3x with x^3, this is an exact example of a lack of relational understanding. They may able to get the right answers when they just learned with instrumental understanding and without really understanding the concept. However, after a period of time, they will forget how to do it and then they will get confused, mess up with other concepts. While I am reading through this article I have to stop a few times to ponder and reflect upon the other examples like the method of teaching fraction multiplication and the circumference of a circle. I realize how severely missing the relational understanding from mathematical teaching.