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