Assignment 3 Final Version

 Assignment 3 Final Version

Course Reflection

 In this class, we looked into how to teach math to high school students and how to become a better teacher. We looked into many papers on pedagogy and we went through the government websites to become familiar with the provincial curriculum and methods of teaching mathematics. I enjoyed doing all the math puzzles given in the class. I think the process of figuring something out is essential to learning. We had a chance to explore different ways to integrate technologies into our teaching to achieve deeper engagement from students. Moreover, we studied the cross-disciplinary approaches incorporating mathematics with subjects such as music, art, dancing, poetry, etc. It was mind-boggling to see different ideas my peers came up with in the first assignment. We also did small micro-teaching and experienced lesson and unit planning. This is a good way as a preview of what is going to be like if we are becoming a teacher in the future. I think whatever we have done in this class will become an ongoing process, with more and more experience and knowledge accumulated. I am very grateful for having Susan who is a wonderful instructor! She made this (the whole covid situation) easy for us. I took quite a lot from this class and from Susan. So, lastly, I would like to hope everyone a safe Christmas and a happy new year! 

Math Food

 

Assignment 3 - Unit Plan Draft

Assignment 3 - Unit Plan

Arbitrary vs. Necessary

 

In this reading, the author presented a very interesting way of viewing the mathematics curriculum. I wasn’t quite sure at the beginning what he meant by “If I’m having to remember…, then I’m not working on mathematics”. But after reading the article, this made total sense. The author made a distinction between what he considered as ‘arbitrary’ and ‘necessary’ knowledge. When something is categorized as arbitrary, it is often something like names and conventions which for most of the time not self-explanatory. There aren’t clear reasons why something has to be called by a particular name and the student just needs to accept this name rather than question or argue about it. Hence, a knowledge that is arbitrary would require the teacher to inform the students and later memorized by the students.

On the other hand, if something is considered ‘necessary’, it is likely something dependent upon the awareness from the student. Unlike arbitrary knowledge that has to be taken as it is, necessary knowledge is something that one can work out by themselves. This requires students to make connections and make sense of what they have learned. Instead of memorizing, students need to reason. However, there are chances that during the teacher demonstrating some math content that is necessary, students mistakenly take them as arbitrary. Students mimic and memorized the steps from the teacher’s demonstration and use them as facts. This is what the author called the ‘received wisdom’. This is a risky thing to do as one can memorize things (or the context) wrongly and have no possible awareness of knowing.

As math teachers, we need to have a good balance of teaching arbitrary and necessary knowledge. We want the students to have the ability to generate new knowledge from adopted conventions. Teacher’s role is to inform students with enough amount of arbitrary knowledge and bring students’ awareness in necessary knowledge so that they don’t rely on memorization. For example, if I were to teach trigonometry, instead of asking them to memorize special trig values, I would ask them to know how to get these values using special triangles as follow.

Self-Test and Reflection

(1) Summary, TPI results

Summary of results

Among the five perspectives: Transmission, Apprenticeship, Developmental, Nurturing, and Social Reform, I scored highest on Developmental and lowest in Social Reform. Also, all my scores are above 30 and below 36. All five perspectives are not differed too much from each other. The profile looks more “flat” rather than “step-like”. According to the TPI website, Dominant perspective is “Developmental” and the recessive perspective is “Social Reform”. According to TPI website “Nurturing” perspective is a Back-up. In terms of the sub-scores of Belief, Action, and Intention according to TPI website, there is high internal consistency in Apprenticeship where Beliefs, Intentions, and Actions are all aligned with each other. There is an inconsistency among the sub-scores in Transmission. My highest intention sub score occurs at the dominant perspective.

(2) What the tests say (and) Surprises

First, let us look at each of the so-called five perspectives. The red underlined parts are the ones (but not limited to) that lack clear definitions or that cannot have clear definitions. These may be fuzzy terms that are dependent on individual, topic of subject under study, and so on. The relationship between explanation of the five perspectives and the questions in the test, the formula to compute the test scores/profile score from the participant answers are lacking from the TPI website.

Tr = (Belief = 14) + (Intentions = 8) + (Actions = 12) = 34

Mathematics is developing with respect to experiences, situations, point of views, and so on. A math teacher may be a “master” (not the masters’ degree) in a certain topic matter when looking at that matter from some points of views. In the other points of views on this same topic, that teacher may not be a master.  Students have right to learn a topic matter from different points of views, not just those points of views the teacher is mastering at.  The integer number scores of 14 for belief, 8 for intention and 12 for action (also for the other four perspectives) are a little bit of surprise not only because these three terms are fuzzy but also because the definitions along with formulas to compute these numbers are not declared, let alone discussion among diversity.

              Ap = (B =11 ) + (I =11) + (A =11) = 33

A surprise is when I read “accessible language” which I always try to do. However many of the educational (research) papers we read including this TPI has jargons and long sentences that could make them inaccessible to many citizens. That means, there is a lack of participation by the people for which educational papers are written. We remember that education is for all. 

Dv = (B =11 ) + (I =13) + (A =12) = 36       

This part is the one that contains the least amount of confusion. A surprise is that this score is the highest among the five perspectives. 

Nu = (B =13 ) + (I =12) + (A =10) = 35

“Absolute achievement” is a surprising term. What is “absolute achievement”?. If there is such a definition by someone and all of us have to follow, then there is a big “Why”? Apart from that, the rest of the content is understandable. 

Sr = (B =11 ) + (I =10) + (A = 9)  = 30

I was brought up in the society in which peace and cooperation are considered important things. When I reached college and university, I read from more diverse resources such as alternative media, news directly from people who went to the place of the event of interest,  and listen to people of diversity. I have come to understand that science (including math) that we know and are learning can be used for both “good” (from human convenience/desire point of view) and “destructions” such as war; atomic, chemical, biological and genetic change weapons; climate change, etc. So the things such as “value” embedded in the “text” are confusing. The term “learners are positioned” is another confusing term. Learners have the right to position themselves with respect to a “text”. They would learn what is said at first. Then they could position themselves to ask questions about validity and reason of that “text” and who made that “text” and why and how that person had the right to do that “text” and so on. Learners can question on modifications, generalization, alternative to that “text”. Learners can position themselves as inquirers.  

(3) Do I agree that the TPI results are currently  my perspective on teaching? Any interesting questions?

(3a) The design of the test, the score number calculations in the profile sheet (result of completing the test) may carry logical explanation or explanation based on diverse and regularly updated solid evidences. I have no access to these information yet.

(3b) The test is made of constructed questions. Each question can be answered by selecting exactly one of the given constructed answers. The designed form of answers are in ordinal scale from SD = strongly disagree to SA = strongly agree for the questions in part two, and in frequency scale from N = never to A = always on part three and four. The possible variations of the answers with respect to the time and subject topic matter is not given a chance.

There are some words such as “not enough” (part two), “expert” (part two), “a lot of information” (part three) that may have different definitions depending on the individual. There is an “and” between “complexity” and “inter-relationship” for undefined “things” on part three question No.28. The answer could depend on the “thing” under interest.

There is another “and” between “accuracy” and “in allotted time” w.r.t. completion of “required content” on part four question No.31. This design may be on purpose, only that other options such as “complete in allotted time but not accurately due to inclusion of external knowledge or modification of some items in textbook due to new publications” are missing.

There seems to be no information on if the design of the test has been modified over the time for some reasons. All in all, this test may be a good first starter which is open to be monitored by citizens (because education is for all) regularly and updated whenever necessary on participation of diversity. And personal information are solely owned by the respective person. 

(3c) The design of the questions, calculation methods to obtain the integer scores are unexplained. With the lack of explanations, the statistical results may not carry the weight. 

Is the highest score in “developmental perspective” due to the least confusion? The answer to this question may be heavy on bias. Even though I do not divide my teaching philosophy into five such perspectives, my opinion is that I do not put myself into making importance-level assignment to these five perspectives. If I happen to put more weight on one at one point, then at the next point I might be putting more weight on the other. If I am forced to compare, I would likely to say equality across these five perspectives and within each perspective (B,I,A), and I would not be able to explain what that “equality” is. 

It could be that this TPI (if not used for judgement) may work with greater validity for teachers of some subjects but not so much for mathematics. In math, we investigate in depth and width, and from different points of views, as well as using logical reasoning. Consistent definitions which are regularly improved as human maturity grows is a characteristic of math. Therefore, to be honest, the test results do not fully represent my current perspective on teaching. It may represent some (this “some” is not according to these “five perspectives”) of my perspectives fairly but certainly not all. 

Thinking about math textbooks

My responses to the textbook content (texts and images) are not entirely in the model constructed in the paper by Wagner & Herbel-Eisenmann. In my view, the primary purpose of the text and image selected by the authors in a section of mathematics textbook is usually to demonstrate the theory in that particular section. For example, pre-calculus textbook printed by McGraw-Hill Ryerson in 2012 has the following text and image in the section of logarithmic and exponential equations.

Using this equation, students see the application of logarithmic functions and practise using them. After playing the position of learners, students and teachers can play the positions of the inquirers. 

A: Directly related to the text and image

(1) How this equation was developed?
(2) When and where this equation was developed?
(3) What mammals were used in the research that arrived at this equation?
(4) Is it approximation over some types of dogs (particularly white dogs from the image)?
(5) What is the precision in relation with the range of m-values?
(6) Would this equation be still valid in the future or in the regions outside of the research area?
(7) Why they choose white dog for the image?

B: Expanding the inquiry

(1) Would that kind of research be done to develop other equations?
(2) Can we approximate the total amount of food per day needed to feed the domestic dogs all over the world that fulfil the calories requirement?
(3) Can we approximate how much money is spent by the dog owners per day buying this magnitude of food per day?
(4) What about the production, from farm-to-mouth processes, business activities involved in making this amount of food available to all domestic dogs?
(5) With respect to the processes and materials involved, what are the environmental consequences?
(6) What about the social conflict or inequality that could emerge because we have students who own dogs and students who do not have dog in class? Or, could this be an opportunity for making a peaceful constructive discussion for our future?
(7) Can we do this kind of math-work for human being with all such inquiries repeated?
(8) Then, how do we compare the people who eat more and the people who eat less with respect to any context of our choice?

The role of teacher is helping expand the idea contained in the ‘texts and images’ not only in the textbook but also from external resources relevant to the topic under study. Teacher listens to the way students view the content and make productive and meaningful discussion possible. Students change position between learner and inquirers.

The diversity in the students in the class would make diversity of such inquiry. Each student brings to the class the culture, knowledge and environment in which the student grows up. Environmentally conscious student would like to inquiry about the relationship between the text/image and environmental matters. Culturally sensitive student would like to ask relationship to the breed of mammal. 

Students who own a dog and students who do not own a dog could have different inquiry or debate over certain point of view. The location of the school can also play a role. The inquiry on ‘where’ the research was done in developing the equation in the text tells us about this.

Let us look at the example in the paper:

This example seems to be demonstrating the application of linear functions. Outside of this scope there is again a large world for inquiry. We can make similar questions as above. Due to two equations involved we can add inquiries with respect to gender.

This time it is related to human being. The image used is of a male can be of a certain ethnic group. This image can create diverse cultural discussions. The accompanying text does not say if these two equations are for all ethnic groups or for some. In addition to the purely mathematical inquiries as above we can add more depending on the experience of teachers and students and level of freedom or comfort they enjoy in public matter discussion.

1) Are these two equations valid only for the ethnic group of the person described in the image?
2) If students think yes to question (1), then more questions arise on the equations for other ethnic groups?
3) What about the geographic, climate, food and life style differences on these equations?
4) If we write the generalized equation as H = a + bF, then what values of ‘a’ and ‘b’ would be advantageous for a particular type of sports?
5) Using more complex formulas and knowledge, will future human genetic engineers, under certain leadership, attempt to modify the genetic design of us? Will that be a disaster in genetic, ecology and entire world? How do we prevent such kind of events?
6) If the ‘a’ and ‘b’ values of certain group of people are significantly different from these two equations, are the people of this particular group abnormal in some sense?
7) Suppose there is a physically handicapped student in the class. Will this study create feeling of inequality? What shall we talk when we arrive at this page in the textbook?

We can also inquiry about authors who make decision on what to include in the textbook.
1) Can the authors of the textbook explain to the public on their selection criteria?
2) Who choose the authors? Do members of the public participate in this? How can a public participation in school textbook design be possible?
3) Can we look at job advertisements or detail descriptions of the works of authors?
4) Are there stories on ‘from thinking and feeling to printing’ processes of actual design of content written by authors of school textbooks that we can read?
5) Do businesses dominate the design of school textbooks? What are the consequences?
6) What is the relationship between authors, textbooks, teachers, teaching resources actually used in class, class discussions, student families, and their future world?

Conclusion

The role of textbook is complex and diverse. Making of textbooks and how they are used are important because of their contributions to future. In general, educators (authors, schools, authorities, teachers inclusive) should be mature, well experienced in their subject matter plus in general knowledge, logical, and diversity-minded persons. Any piece of public work should not be done by a group of people. Diversity-involved discussions and decision making are needed. Diverse, deep, wide, logical and critical thinking should be promoted in class because today students will become leaders, authors, publishers, business owners, politicians, and decision makers in the future. 

Here are some reasons for using a certain content of the text book:
(1) Relevant to the curriculum
(2) Enough amount of resource provided (e.g. time)
(3) No content that could create atmosphere of social inequality, below is an example from an exercise problem in a textbook. This is the type of content to avoid using in teaching

(4) Content offers critical thinking, generalization, and discussion for the betterment of the future
(5) No better alternative available  

Reflection on the group microteaching

Our topic for the microteaching is introduction on slope. The overall structure is organized. We started with a hook relating the audience's experience in hiking. We then started the explorative activity and the formal "lecture" to sum up the learning outcomes from the activity. Since the lesson is designed for grade 10 students and our audience is actually university students with math background, we should have remove some repeated examples during demonstration so that we don't need to rush in the end. However, we did time ourselves and we ended up finishing this lesson in 18 minutes (there was a technical glitch at the beginning where none of us can share the screen). We prepared some additional worksheets but we didn't get to present those because of limitation in time. The example we used to illustrate slope as steepness of a hill is very visual and easy to understand. This might encourage student to pay more attention to the lesson because they can relate themselves to the real life example. During the lesson we tried to have the audience engaged by constantly asking questions and respond to their answers. Overall, we had good rapport with the audience and all of as were confident when we were presenting. Different from teaching individually, we learned that co-teachings need not only the quality of content delivered but also the cooperation between the co-teachers on how to deliver the content so that students' learning experience can be enhanced.    

The Scales Problem

 A market vendor sells dried cooking herbs in whole-number amounts from 1 to 40 grams. The vendor has an old-fashioned two pan weigh scale, and has exactly four weights of different amounts that allow them to weigh out any of these amounts of herbs -- without using the herbs or any other object as an auxiliary weight. What are the values of the four weights? Are there several correct solutions? How could you extend this puzzle to help your students understand the mathematics more deeply?

Solution content:

1) Formulation of an extended problem of generating a range of positive integer value
2) A special case of two pan weight scale
2a) Attempt using base-two binary number system
2b) Attempt using base-three number system with modification
2c) A construction procedure to find alternative solution

Group Microteaching Lesson Plan

 

The Giant Soup Can Problem

 

Write the given information: 

Look at the photo of the fallen water tank with the bike. We choose the dimension of the water tank (height or diameter) that would produce the minimum error. The dimension chosen should be distorted the least, and is parallel to a bike dimension (for example, height). This dimension and bike dimension should be close to each other on the camera line of sight. 

Suppose we choose the diameter. From photo we measure and obtain D/(h_b) which is 
(Diameter of the tank)/(Height of the bike) value using a ruler. Then, following computations estimate diameter, height and volume of water tank.

We can find, or construct and agree on, what are acceptable definitions of average house fire and required water amount (volume) to put out the fire. Let (V_ f) be required water volume to put out average house fire. Then,

Expanding the idea: Let us work on the same water tank with student birds in class. Given the photo of the fallen and little bit distorted tank with a bike, how do we estimate the dimensions and volume of the water tank?

Some students would want to organize the information we already have first.

 What we have:
1) The water tanks have the exact proportion of the Campbell’s soup can
2) The photo showing the fallen distorted tank and a bike

A student asks what is proportion? Another student answers, “It is diameter to length ratio”. When a key moment arrives, teacher steps in and reminds that we need clear statement written in completion not to make confusion. The variable of interest is volume of cylindrical water tank. If V is volume of water tank, v is the volume of soup can, then

It sounds like teacher forgets something. A student in the class has parents working in sheet metal business. This student points out that we are forgetting the thickness of the shell material used in construction of water tank and soup can. Someone recommends that we add assumption.

 Assumption 1: Thicknesses of the materials are negligible compared to the volume.

 What is next? Students shout out we need that proportion. Some students argue that we do not have a road map. We have not got a plan yet. Then one says we are planning now. Someone quietly listening now suggests that it is all good because we are in an adventure. We argue, point out weak point, but everything is peaceful. Finally a constructive idea is reached.

 We would copy, cut and match the water tank photo and the actual picture of the soup can. Another debate erupts on “what to match and why?” The tank in the photo looks distorted. A student asks if we can match the size of the letters because these letters seem to be the most intact in the photo. Another student agrees but adds assumption.

Assumption 2: The proportionality includes letters on the label. That is,

Students use school computer to do some image works. They use zoom in and out while keeping the proportionality. They compare, and finally they manage to get the letters on the tank and soup can nearly the same size. Therefore, the original soup can picture serves as original water tank alongside the damaged fallen water tank with the bike.

If the assumption (2) is true, this picture surprises the teacher. The water tank in the photo is distorted significantly. The answer quietly calculated by the teacher before seems to be not realistic now. This would become another key point that ‘looking from different points of view’ is beneficial. Teacher decides to discuss this after the project.

Students take measurements on the final picture and compute the ratio H/L and D/L. Now we need more information. We need the actual length L of the bike. Students arrive at a mid-point conclusion that we need the length of the bike to get water tank dimensions. Therefore, the class adds a new assumption.

Assumption 3: We have the length L of the bike.

Students then do the computations:

We do not even need the actual dimensions of the soup can. We also have a student whose parents work in some engineering design jobs. This student remembers that our assumptions may be incorrect and we need to inspect. When the student points that out, the student is not very sure what exactly that is. Teacher catches another key moment to explain what is in the student mind.

Sensitivity analysis:
If a value in the computation is actually different from what we assumed, then the final result can change. What is the effect of a certain amount of variation in data on the final result?

Before the teacher bird opens her mouth to confess her prior solution and its weak points, student birds demand for explanation why we need so-called sensitivity. Instead of answering directly, teacher asks if they want their firefighting water tank to be smaller than the necessary size? This sparks another round of discussion for importance of responsible reporting of computational results in real life. This class project produces some important points:

• Assumptions must be made after reasonable amount of unbiased inspection
• Statements (including reporting) should be produced in completion to avoid misunderstanding
• When seeking a solution (in this case minimum error estimate), past experience and different points of views or methods should be applied
• Need of experiments, exploration, and learning of skills and reasoning required by the problem at hand (in this problem, knowing that two images need to be at the same state and actually making it happen)
•  Possible weak points (for example due to assumptions) of each method should be discussed
  
• Sensitivity analysis on all assumed values should be done
• Responsible reporting

Additionally, if we do not have a means to manipulate the pictures (advantage of technology), we would still need the actual dimensions (letter size also) of the soup can. Finally everyone agrees the importance of sharing of diverse views, methods and knowledge; and need to continue learning to gain greater maturity (experience). It is almost 7 pm. It is decided that all birds need a rest. 

1

Flow

 From my experience, when people see high interest, high value matching their world view, truth that worth knowing, challenge that matches and tests their skills in doing something they are into it. They are willing to spend time and energy into it. According to TedTalk, it is not that income and living standard, but it is something meaningful and worth doing for the person. It is ecstasy or a state of the person’s mind that transports the person into alternative reality. It is a completely engaging process in which the person sees creating a new thing. According to Peter Liljedahl’s presentation at SFU (Engaging students, understanding flow, 2015), there are nine items involved in the state of ‘flow’ in which a person is completely engaged and delighted. These are:

1. There is clear goal every step

2. There is immediate feedback on one’s action

3. There is balance between challenge and skill

4. Attention is focused on one’s action

5. Distractions are excluded from consciousness

6. There is no worries of failure

7. Self-consciousness disappears

8. The sense of time becomes distorted

9. The activity becomes satisfying in its own right

In teaching math, teachers can take part in dialogue with students on the world we live in, the society humans have built out of natural resources and the planet, the villages, towns, trades, transportations, necessary skills needed to contribute to work and earn, and sports, arts and so on. We will see that math can be found in almost every component. That way, teachers can establish interests in students. 

Students have their family and friend circles or communities. They grow up in certain neighbourhoods in which there are also works. Therefore, students may already have some interests. Or students can be interested in some other works they found in movies, the Internet or from someone. Teachers need to be well rounded, mature, or able to learn the subject matter of student interest in relatively shorter time. 

Teachers then ask the topics of interest of high value of the students. After getting a list of topics of interests, students are asked to rate their interest on a scale to each of the topics. That creates an interest level (1-5) table like this.

Teachers can declare that there would be a certain number of projects (3 or 4) to be undertaken in the semester. So the class would use the table and vote for the project topics. On each project teachers carefully plan the start point, possible process and ending so that the level of challenge is appropriate to the class, and design the project question based on the students’ inputs. Teachers also explain to the class about the concept of ‘flow’ ahead. Students will participate in keeping the project process within their challenge boundary without losing goal in each step. Students participation is important to let them keep themselves in ‘flow’.

Or the class can be divided into two or more subgroups working on different projects of interest to the sub-groups. This sub-group method could make the project and level of challenge more suitable to each sub-group. 

It is also important for teachers to discuss with the students that we as humans living on the planet earth (which is not our design and construction) can face any event in our life. Some events we have gone through in our life can be not our interest but we have to face it. So it is necessary to raise the issue that students need to be able to solve the problems of not only their interest, but also other problems.

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.