Battleground Schools Summary

In reading this article I have formed several thoughts. Firstly the relation of mathematics education to politics. I find this particularly unsettling... as I have always viewed mathematics as transcendent beyond all corporeal concepts including politics, religion, culture, etc. Perhaps this view point regarding mathematics is apt it is apparently not so apt when applied to mathematics education. Throughout the past century educators have grappled between traditionalist approaches to mathematics education and progress approaches to mathematics education.

In regards to the New Math era it seems that the mantra of the epoch was to model all mathematics education in the style of Carl Friedrich Gauss, whose book Disquisitiones Arithmeticae set the tone and style for advanced mathematics textbooks until the modern age, which is to present mathematics neatly as beginning with a set of axioms, then proceeding to state and prove simple results known as propositions, then build up various theorems using propositions and lemmas proved, and a few pithy comments spread throughout to keep the reader in check. Modern textbooks would also have a few exercises in each chapter. This is strongly influenced by Gauss's personality as a perfectionist. Gauss was famous for destroying the scratch work he did before proving a result as to show the world only the 'perfect' mathematics. This is counter to how mathematics is actually done or learned however, and I believe this attributed to the failure of the New Math program.

Another key factor leading to the collapse of the New Math program is the 'golden age' effect. In essence, it comes to policy makers who are usually very gifted and skilled in the subject area, remembering that when they went through the lower grades of school that they were not challenged. They then inferred that adding a few 'simple' additions to the mix that 'anyone' can understand and learn would greatly improve education overall. These assumptions are not valid of course, as they usually use themselves (probably in the top 1-3% of the population) as the 'norm'.

In the current curricular reforms the battle between traditionalist and progress views again associates inflammatory politics into mathematics education. I personally feel that quantitative literacy is key to the longevity of a liberal democracy. As such it is certain that high quality mathematics education to all adults of society is crucial to the continued success of our democratic society. In particular all students need to be taught to not only carry out rote exercises but to interpret mathematical data, models, and statistics as to inform themselves about the current state of affairs in the economy and politics. Thus the conception that mathematics ought to be taught only to a few academic elites and then disseminated to the masses as needed is as outdated as the concept that only a few people ought to be literate and everyone else can get their information from those educated elites. Thus, reform in mathematics education is urgently needed.

Personal Summary of Assignment 1

In this assignment I found that teachers seem to not want to deal with behavioral issues in class. This to me seems disagreeable but understandable. However personally I feel that teachers have a responsibility to promote an environment where behavioral issues can be addressed and corrected rather than be swept under the rug and hope that 'delinquent' children don't come back the following year. The teachers interviewed seemed to not wish to plan out lessons as rigidly as one might thought. I believe this has merit too, since quite often elaborate lesson plans get discarded during a lesson plan as the class takes an unexpected turn. Planning for maximum flexibility seems more effective than having a fixed plan in mind. Finally the teachers opinion of variable standards is something I need to ponder about more still, hopefully after I have a better conception of the structure of mathematics education at the secondary level.

Summary of Assignment 1

We interviewed two teachers and ten students. The first was Dr. Cailin Lucus and the other was Mr. Ahn, a math teacher from Fraser Heights Secondary who was interviewed via email. Most of the students interviewed were students that Paul tutors from Riverside Secondary and Centennial Secondary.

From this experience I have surmised that a principal challenge for math teachers is typical school politics, mastering the curriculum and course material, challenges in teaching different grades, and lesson planning. One of the main points identified regarding school politics is challenges over grades, particularly in the 90+ range and the ~50 range. This is understandable as top students strive to get as high a score as possible for scholarship and post-secondary opportunities while struggling students try to pass the course. This usually requires some tactical diplomacy between the school administration, the teacher, and the parents. Both teachers believe that knowing the curriculum and teaching in an appropriate manner to the curriculum is key to success. Regarding lesson planning, it seems that it is important to have a solid repertoire of lesson plans built up initially and then adjust as needed as time progresses. Both teachers find teaching younger grades more challenging due to a higher number of weaker students and more behavioral problems.

Most of the students interviewed believed that mathematics is something that is quite accessible to everyone and not just the academic elite. Most seem to have at best a lukewarm attitude towards mathematics and most only take it because it is a requirement for higher education or graduation. However, almost all of the students believe that mathematics is important and useful.

A few points of summary include that new teachers should plan carefully and as they become more experienced can use terser forms of lesson plans, such as checklists. Also, classroom management is more important for younger grades than necessarily delivering the best course possible. Finally, a teacher needs to know more than the textbook and in particular ways to engage students in class to increase interest.

Reflection on Robinson's Article

I found this article somewhat an attempt to preach to the choir. The techniques she described are ones I am already aware of and I am always wfeary to hear learning strategies from people who themselves have a dubious background in mathematics. Any accomplished mathematician would see manifestly the value of problem solving in all facets of mathematics pedagogy. However she did serve as a good reminder that a goal oriented classroom is important and to stay in touch with the students. In her case, she was not in touch with her school's final examination nor her students' preparedness for it. In my own teaching I like to mention the final goal as soon as possible and unite that class behind that goal.

My Two Most Memorable Teachers

My two most memorable teachers are Professor Greg Martin and another mathematics professor at UBC who do due to him/her being memorable for negative reasons shall remain unnamed.

Professor Martin is memorable to me because he always gave the impression that he truly and whole-heartedly loves the subject he is teaching and mathematics in general. He always integrated humor into his lessons and had a very student oriented approach to teaching. He made sure he put himself in our shoes and presented the material in such a way that is accessible to students seeing the material for the first time. He emphasized on problem solving techniques all the while maintaining that computational accuracy and effective problem solving are inseparable. He also had an extremely open door policy where he invites you to discuss not only material related to the course he is teaching but all mathematics in general. His enthusiasm in both mathematics and teaching have rubbed off on me and have motivated me to become just as great a teacher.

The other professor who shall remain unnamed is memorable because of the extreme coldness and acrimony he displayed to the class before really even meeting the group. His/her first comment to the class was quite boldly "hi, I am Professor So and So, and 60% of you will fail this course." He/she said it with such conviction that it was doomed to become the truth (in reality, only 54% of the class failed). Though this professor may have been speaking the truth based on statistical evidence, such a negative opening statement set the class up to fail.

In looking at the above two examples there are two extreme examples. Both individuals are truly subject matter experts in mathematics and yet due to their teaching styles and the image they project have become vastly different instructors. I hope to learn from both the enthusiasm of Professor Martin as well as the apathy of the other individual as guidelines for my own development.

First Microteaching Reflection

Today I taught 3 people about the Canadian Coat of Arms. Some of the comments I received include that I read off my lesson plan too much, that the diagrams I provided were too small, there were too many consecutive minutes of lecturing, and that the summary was too brief. I have noticed these issues myself and have considered measures to correct them in the future. Obviously to correct reading off the lesson plan too much I must endeavor to study the lesson material more in depth. I can easily print off a larger version of the coat of arms off of the Heritage Canada website, and perhaps even blow up the diagrams I already had. To prevent too many consecutive minutes of lecturing I can interject questions in between each portion of the lesson. Finally I can easily give a summary of the lesson learned at the end.

Lesson Plan - Canadian Coat of Arms

LESSON PLAN - CANADIAN COAT OF ARMS

Bridge: Can you name a symbol that represents all Canadians? If so, why do you believe this is a suitable symbol for Canada?

Teaching Objective: Nil

Learning Objective: Students will be able to identify this prominent Canadian symbol and all of its constituent components.

Pre-Test: Just ask what students know of the Canadian Coat of Arms (not knowing what it is, know that it has a crown on top, know that it has the Union Jack on it, etc.)

Participatory: All students will be given a piece of the Coat of Arms and at the end of the class will be asked to identify their piece on a full diagram of the Coat of Arms, and give an explanation to the meaning of the piece.

Post-Test: See above.

Summary: This lesson identifies an important Canadian symbol that appears ubiquitously on most Federal buildings and is a prominent symbol in the military. I hope that you have gained a bit more knowledge about your Canadian heritage.

Reflection on the Skemp Article

I find that this article advocates a view point but does not necessarily use the best examples. For example the article stresses on computing the area of a rectangle as an instance of instrumental teaching. It offers however no explanation on why the area of a rectangle is base times height. There is no obvious answer to this; since area from a mathematical stand point is merely the Lebesgue measure on the plane. Of course given the ubiquity of the concept of area across all cultures old and new indicates that something more basic is at play. This of course is that the concept of area is the generalization of the number of objects in an array, where an array of 5 rows and 7 columns would have 35 squares, similar to how a rectangle of length 5 and width 7 will have area 35.