Geometric/Numerical Puzzle

Rotational symmetry of order n means the object can be rotated about the point of symmetry by 360/n degrees and the result is the same as original in geometry. Circle has a rotational symmetry of infinite order with respect to its centre. That is, rotation of the circle about its centre by any angle does not change its geometry. Two diametrically opposite points on the circumference divide the circumference into two arcs of equal lengths. Thirty equally spaced points on the circumference make 30 equal arcs out of the circumference. From the point (7) one travels 15 arcs (half the circumference) to point (22) which is half way around. Therefore points (7) and (22) are diametrically opposite to each other.

The process to the solution is based on these two facts about the circle:
1) Circle has rotational symmetry of infinite order.
2) The same angles at the centre correspond to the arcs of the same length.

An extended puzzle: A farmer has fencing material of length 600 meters. The farmer mistakenly assume that circumference of a circle of radius r is 6r. He starts building the circular fence with radius 100 meters, thinking that he would get a circular enclosure. After laying just over 300 meters of fencing material, he realizes that he is wrong. He manages to close the fence by laying the last part straight. Does the farmer get the maximum enclosed area?

 The key to this question is to know what region on the flat plane has maximum area to perimeter ratio. The circle has the maximum area to perimeter ratio. The farmer finishes the enclosed area of the following shape.

What he gets is not circular region. If he correctly calculates the radius for the circle with 600 m perimeter, then he would get

With this radius he would get a circular area larger than the one in above diagram.

Regarding the solution: In our life when we face with problems some possibilities are:
1) There is no solution.
2) There is one solution.
3) There are finitely many solutions.
4) There are infinitely many solutions.
5) We know there is a solution. However, we are unable to compute exact solution due to our insufficient computing ability (capacity) or due to resource limitation. We can find approximate solution(s).
6) We can simplify the problem to a simpler form. However we cannot get better for now.
7) We do not know if the problem is solvable (need more knowledge).

The number of solutions (including zero) can be due to the constraints. For example, in the problem: x+y = 5 with both x and y are real integers of at least three, there is no solution. But if we can take x and y any positive integers of at least two, then there are two solutions:
(x,y) = (2,3),(3,2). If x and y can be any integers, then there are infinitely many solutions. Students should be allowed to experience a variety of solution-situations.

A puzzle (problem) truly geometric: A puzzle which can be solved by knowing the properties of the geometric figures involved. 

The new BC curriculum & secondary math course pathways structure

• Two things that were new to me or surprised me from the curriculum orientation guide and/or glossary of new terms
  
  First thing that I feel surprised is that “thinking” is listed as one of the core competencies. When I was in school, there was not an emphasis on training how students think. Even in subject like mathematics, I did not see any implementations that were dedicated toward training students to think critically and creatively. Normally math classes were taught in the form of a lecture, which was a one way communication from teacher to students. Students mimicked how teacher solve problems and learn the technique for solving problems. There were rarely any debates between teachers and students on concepts, instead, student would trust their teacher entirely and follow what the teacher did. There were not much creative nor critical thinking. I see the importance in teaching students to think but thinking itself is a very abstract object which is hard to grasp as well as to measure. I glad to see that this is properly addressed in nowadays curriculum and I would like to see how we can follow up with this.
  
  The second thing I read which was not seen during my school time was the use of inquiry-based approaches. As I mentioned, in a mathematics classroom, the method of delivering knowledge was through lecture. Usually after learning in class, we practiced our skills by the assigned homework and used the way we were taught in class. And the traditional way for assessment were tests and exams. Talking from my own experience, I had never done a mathematical project nor an open-ended small research problem in secondary school. All of what we learned is for the purpose of passing exams but not for applying them in the real life situations. I have heard a saying that “education is the kindling of the flame not a filling of a vessel”. It is very important for educators to elicit the interest and curiosity from students so that they can take ownerships of their learning. Inquiry-based approach is one good way to actually implement creative and critical thinking skills which is lacked in traditional approach.
  
  
• My own schematic chart of possible pathways in the courses of the BC Math curriculum from Grade 8 - 12.
  
   
  
  
  

Micro-Teaching lesson plan with self assessment

 

The three curricula that all schools teach

 

 It was a pleasure reading this article. I agree with many things Eisner discussed. Usually we only pay attention on what explicitly schools teach young people because at the end students’ performances are assessed based on these explicit curricula (i.e. subjects taken). I personally never thought about what schools implicitly teach. Since children spent most of their childhood in schools, every element and aspect of school environment should be counted for shaping the behavior of young people. In fact these things implicitly taught might have even bigger and longer influence on a child life into adulthood compared to academic subjects. These implicit curricula usually involved in preparing and habituating the students to cope with lives after or outside of a school setting.

Equally important to what schools teach is what schools fail to teach. One thing out of many is the development of cognitive processes. In subjects like mathematics, one of the goals is to improve student critical and logical thinking abilities. However, not a lot of things explicitly taught in school reflect these objectives. There is no guidelines for implementation nor sections in a subject that specially dedicate towards the training of how students think. Teachers usually teach out of a habit and can often neglect things that are “outside” of the subject content. I think during teachers’ education, we should learn how to train students to think using the means from the subject area as well as the physiological and psychology factors behind it.    

Curriculum often means the subject explicitly taught in school. However, one might consider schooling as a package that comes with all the three curricula that Eisner pointed out. It is not possible to only learn the explicit curricula when a student is raised in a school environment. In my opinion, school board should address the implicit curriculum as well as the null curriculum which can be addressed as an objective or goal to achieve in teaching. The current BC curriculum in mathematics addresses some of the cognitive thinking skills in the big ideas sections. However, there is no apparent addressment on what Eisner considered as implicit curriculum.   

Microteaching lesson plan

 

Microteaching Topic

 For my microteaching class I will teach my group how to solve a Sudoku puzzle!

The Dishes Problem

By learning and working out these ancient puzzles (questions) and mathematics history from diverse cultures without using today algebra, we and students learn word techniques without symbolic algebra, as well as some information about the lives in ancient times. Doing these problems in a way ancient people would do gives imaginative and writing skills needed due to restriction from using symbols. Since it is a different way of reaching the same solution, it gives a new enjoyment. Word problem or puzzle story with images, it does not matter. We get some knowledge from both. 

Battleground Schools: Discussion and reflection

 Mathematical education ideologies can be characterized in two polarities, namely progressive and conservative. From reading this article, I found that the development in mathematical education curriculums in the US lines closely with its historical and political events. The dominance of the  different two polarities depends on what was happening globally at that time. 

When reading the table for comparison between conservative and progressive teaching, I was surprised that throughout my education in the past 20 years, I was, especially in mathematics, taught more from  the conservative method than progressive. I did my elementary education in China and I was told before coming to Canada that I will find Canadian education very different from Chinese. I agree with that up to certain points. Here in Canada, we get less homework load and we don’t value marks from examinations as much as back in China. But in terms of the goals of math learning, nature of student work, and many more areas of interest, I find little difference between the two cultures. And during my time in  secondary and postsecondary education I did here, I found the teaching styles reflect more of conservative rather than progressive. 

Another thing I found interesting was that during the progressive reform, Dewey proposed the idea of doing mathematics, with experimentation and inquiry. Even though the process can be hard to control, messy and unsettling, it is thought to produce “scientific and democratic thinkers rather than rigidly obedient rule-followers. So, does this imply that the “obedient rule followers” cannot be “scientific and democratic thinkers”. With the belittlement of valuing precision and correctness, does this imply solving a mathematical question correctly is less important than the process of “exploring”. My concern is that the process of exploring can be hard to assess and grading such processes can be subjective. Therefore, how to standardize and make it practical should also be addressed. 

In the New Math period, I was impressed about the attempt math educators made to “create a unified, logical, highly abstract algebraic structure”. I actually like the idea of integrating some areas from modern mathematics into school math curriculum. That is because subjects like set theory, abstract algebra, and etc. don't require a high level of calculation skill. Instead, immersing students with the idea and broadening their view in mathematics can be helpful in developing thinking skills. This transition needed a longer period of time to accommodate “local conditions, cultures, or educational traditions”. However, with lack of persistence and because of its high demand, New Math programs started to fade after the 1970s. I am wondering, since not all students are suitable for the new math program, can school make it as an elective course made for students who are interested and willing to spend effort in learning it. I like the New Math program but I hardly agree with their view that “every student was a potential future rocket scientist.  

Reflection on Art/Math presentations

I was impressed by the quality in all the presentations. I have never though that there are so many different connection one can make between mathematics and art. People really carried the topic above and beyond.  It is for sure different presenting online compared to presenting in a classroom, yet all the groups made good effort in preparation and made it happen. Also, it is such an experience working with our peers virtually throughout this project. This will prepare us well in the future teaching career in case face-to-face instructions are not available. I look forward to seeing more awesome projects in the rest of the semester. 

Math and art presentation slides

 For those who are interested, here is the link to our presentation slides. 

EDCP 342 Math/Art Presentation