Math and Sound Teacher Notes

Grade 12

Trigonometric Identities

Paul, Stanley, Erwin, and Gigi

Purpose: To show the trigonometric identity:
sin(2πf1t)+sin(2πf2t)=2 cos(2π (f1-f2)t/2)sin(2π (f1-f2)t/2)
in an auditory manner.

Students will hypothesis, predict and test which part of the wave is associated with the pitch and which part is associated with the beat or the volume.

Description of Activities: Students will be using Audacity (a free program) to generate sinusoidal sound waves. Playing two waves together is like playing the sum of the two waves. (This shows the additive property of waves.) Students will be investigating the beat phenomenon and the trigonometric identity: sin(2πf1t)+sin(2πf2t)=2 cos(2π (f1-f2)t/2)sin(2π (f1-f2)t/2) . They will be following their worksheet to guide them through the activity.

Sources: http://en.wikipedia.org/wiki/Beat_(acoustics)

Estimated time: 1-1.5 Classes or 1 Class and the rest is due for homework

Students are required to produce a page of answers and supporting graphs. Students will conclude cos(2π (f1-f2)t/2) is the part responsible for the amplitude (and thereby the beating) of the wave. The pitch of the wave is the part sin(2π (f1-f2)t/2) .

Marking Criteria: Students will be marked on the correctness of their answers. Students will also be given a mark on their participation in class out of a 4 point scale worth 25% of their total mark.

1- Did not participate

2- Participated 20% of the time

3- Barely meets expectation for participation

4- Meets expectation for participation

5- Exceeds expectation for participation

Math and Sound Investigation

1) Open up Audacity. Go to Generate : Tone to create a tone also known as a sound wave. You can delete a sound wave by highlighting it and pressing delete. It is not recommended to have more than two sounds waves in your workspace at once. Try generating sound waves with different amplitudes, waveform and frequency.

2) At a fixed point in space, a basic sound wave (pure tone) can be represented by a sinusoidal function. One of the functional representation is Asin(kt).

Fill in the blanks with the following words: amplitude, frequency, volume, pitch

The _________________ of the function represents the loudness or _________________ of the sound.

The _________________ of the wave function represents the _________________ or how high or low the note is.

Copy and paste the above two statements with their answers onto separate file. You will print your answers at the end of class to hand in.

2) Investigate:

Generate a sine wave tone. Choose a frequency (call this n)
Generate another sine wave with frequency n+1.
Play the result.
What does it sound like? Record your answer.

You may hear a pitch change and a beating frequency.
Delete the wave with frequency n+1
Repeat the above and investigate with frequency n+2, n+3, n+10, n+20, n+100, etc
How does the pitch change as the second frequency increases?
How does the rate / frequency of the beat change?
Record your answers.

3) On graph paper, hypothesis what the wave would look like.

4) When you are listening to the waves together, it is like adding two sin waves.
sin2πf1+sin(2πf2) =
What is the trigonometric identity associated with this sum?

5) Hypothesis which part of the wave is associated with the amplitude (the beating) of the wave and which part is associated with the pitch.

6) Clear your workspace. Make two sound waves with different frequencies. Record this frequency. Using your hypothesis in question 5 predict the frequency of the beat and the frequency of the pitch. Use appropriate frequencies in which you can hear the beats.

7) Test your prediction by listening to the waves. Record how many beats you hear in 10 seconds. What are the experimental values of the beats? Is your hypothesis in 5 valid?

8) With the trigonometric identity found in 4, label which part is responsible for the beats and which part is responsible for the pitch of the waveform.

9) Plot out the sum in a graphing program or graphing calculator. You may need to plot twice: choose an appropriate scale for the beat and another for the pitch. Include your plots with your answers to hand in at the end of class.

10) Find two frequencies f1, f2, in which a period of the beat is 10 times larger than the period of the oscillations (part responsible for the pitch). Show your work. Graph the sum of sin2πf1+sin(2πf2) .

Extension Questions:

Go to http://www.mindspring.com/~j.blackstone/dist101.htm. Scroll down to the applet. Play around with the applet and record how the sound changes. Does the perceived pitch change? In other words, if you tried to hum the note does it change as you change the wave?

Division by Zero Reflection

As I am not much of a poet, this exercise was quite difficult for me. It seemed unnecessary to teach mathematics. In particular, I even had the feeling that the audience that I can reach through this approach could be an audience I never wanted to reach in the first place. Of course, this type of thinking (that there are 'undesirable' students) is very detrimental for an educator, I still see little value in using poetry in mathematics. This is in part due to my own belief that poetry and mathematics cannot comfortably co-exist. To quote Paul Dirac: "The aim of science is to make difficult things understandable in a simpler way; the aim of poetry is to state simple things in an incomprehensible way. The two are incompatible." Of course this is an extreme view, but I suppose my philistine understanding of the art of poetry makes it difficult for me to appreciate its beauty and worth, much less advocate mathematics through this medium.

Assignment 3: Analyze a Project Idea and Develop/Extend

Group: Gigi, Paul, Stanley, Erwin

Handouts and Sample Tilings:

Reflection for advantages, disadvantages and modifications from each group member:

Gigi:
Teachers and students will need to know how to separate a line segment into different equal number of parts. I had to look this up in Google but the instructions are not always intuitive. I had to recall how to copy an angle. Not everyone has learned how to copy angels, create a perpendicular bisector to a line.
Another problem we had was with creating the star of David. Students may place the triangle intersections in different spots. Students may have to draw on the copy of the tessellation analyze the different shapes that are used to construct the larger shape. Students may need to draw on the copy of the tessellation to see the symmetry or see how the shape is constructed.

Stanley:
After attempting one tessellation I realized that the concept of 4-fold symmetry is key to the tessellation. What a fantastic way to introduce the concept of symmetry. Can easily expand the project by first asking the relatively intuitive question of whether or not any 2-D object with n-fold symmetry can be used to tessellate the plane(the answer is no when n is infinite, as the only object with infinite number of symmetries is the circle which cannot tessellate the plane). Also with this type of construction it is not so important to enforce 'straight-edge and compass' rules of construction as this type of construction is only 'sacred' because of Greek influence.

Paul:
In terms of evaluating the benefits of this project, I think this project caters to the student's artistic flare and mathematical prowess. The first part (drawing the pattern, describing the pattern in words) appeals to those people inclined towards art, not requiring any math skills (except trying to keep things in proportion). The second part (recreating the repeat shape with straightedge and compass, making minor modifications) caters to the mathematician. It requires visual analysis of the shape to look for patterns that can be recreated using the geometry knowledge that students have thus far. Teaching analytical skills will be helpful not just in math but in so many other areas of life. One weakness of the project is that it could be overwhelming for students who cannot see the pattern that can be recreated with the two simple instruments. Tessellations are covered in the Math 8 IRP. Some of these might be difficult. (I will admit: I got a bit frustrated that I could not recreate the hexagram in the middle.) Of course, this problem could be alleviated by choosing simple tiling patterns for the students. I would not modify this too much, but I would pre-select some not-so-challenging tiling patterns or at least have two sets of patterns (medium and hard). One constraint of using this project in my classroom would be time. I might not know how long to prolong this project or how many class sessions to allocate for in-class work. Overall, I could see this being used in my class to get their creative juices flowing.

Erwin:

I was excited about this project. By mixing art and mathematics this was a nice departure from the regular math problem solving typically done in a class. All throughout the assignment, I was unsure with how far I needed to go if I imagined myself as the student. Some students might scrape only the surface of the project and others may go very deep into much detail. The construction is very mathematical because it doesn't follow the usual math problem. By trying to find an object from basically experiment is more real and connects with the individual. I needed to look back at my geometry math book to remember some simple constructions.

Benefits:
-Gets students that are not particularly excited about math a bit more engaged
-Gets students to apply what they learn to out-of the box problems ike finding minimal parts of tiling
-Assessment of mathematical ability can be made

Disadvantages:
-Students will not achieve the same outcome because the instructions/expectations are not clear enough
-The project may in fact take too long for individuals and it might not be able to be applied to groups of people. The project is based heavily on the discovery of shapes not inpendent steps.
-Harder to evaluate, giving a letter grade on effort of the final presentation.

Modification:
Break up the assignment and assess or evaluate each of the steps. This gives clearer purpose to the assignment and is better for time management of the assignment.

Division By Zero Poem

Division by zero,
Deemed the greatest mortal sin,
Long upheld,
Ever since zero came into the lexicon.
Until whence,
Bernard Riemann graced Earth with his presence,
And discovered,
That there is but one infinity among a myriad,
And naturally,
Zero got paired with infinity, and became a whole.

Division, Zero free write

Zero:
- The symbol for 'nothing'
- Looks like a hole
- Under the map z -> 1/z is sent to the point infinity
- 0 times infinity = 0
- the additive identity in a ring
- obliterates everything under multiplication (even in an arbitrary ring)
- the 'center', versus infinity as 'the limit'
- 0! = 1, because there is only one way to arrange nothing
- nCk = 0 if k > n, because there is no way to select a set of k items from only n < k items

Division:
- Multiplicative inversion
- Given an arbitrary integral domain one can construct a fraction field where multiplication makes sense
- Makes sense in a division ring
- More problematic than multiplication because the integers have no multiplicative inverse in the integers
- Perhaps motivated the advent of rational numbers
- 'Division by 0' makes perfect sense in Riemann sphere

Reflection on Mathematics' Role in Citizenship

In Elaine Simmit's article she discussed the potential problems with associating mathematics as a mere collection of facts; as black and white. Surely no pure mathematician would ever contend with these notions. Mathematics is built on a foundation that EVERY statement can only take on exactly one of two values: true and false. All of mathematical logic is built around this concept and all theorems proven is based on this essential assumption. Essentially, assuming the 'universal' axiom that a statement is something that is either true or false (but not both) and a given set of axioms, all mathematical theorems may be proven. While this is essential to building rigor in mathematics it is problematic in mathematics education. The problem is the lack of emphasis that PURE MATHEMATICS need not translate or apply to REAL LIFE situation. For example, there can be no doubt that a sequence of independent and identically distributed random variables with finite mean converge in mean almost surely, it is NOT without doubt that a given sequence of random variables in real life SATISFY the conditions for the theorem. Thus it is important for us educators to identify the APPLICATION of mathematics in real life situation. This is already strongly advocated in, for example, social studies classrooms where there is a strong emphasis on analysis of current events and to unfold the intricacies of modern society and politics. This is not done in mathematics. When thinking of 'real life' problems in mathematics we usually associate extremely trivial things like finding the correct amount of change. This is tantamount to being able to read road signs as 'reading in real life'. While not a FALSE statement this obviously does not achieve the objective. Students are not challenged and don't learn anything. Instead, students need to be implored to discover more sophisticated applications of mathematics, for example how survey companies collect data, how data collected from the census is analyzed by the government and how does this influence public policy, the soundness of using 'average income' as a barometer to judge the social well-being of a community, etc. These will ultimately produce a more engaged learning environment and produce better, more quantitatively literate citizens.

Reflection on "What If Not" Strategy

The "What If Not" strategy is perhaps what distinguishes a mathematician most from any other type of scholar. By asking students questions constructed from an original problem and therefore exploring variations of the same theme students can be encouraged to engage in the same type of thinking that is so important in the study of mathematics. Because the nature of mathematics allows us to negate any physical restraints (for example, there surely is only one reasonable law of physics, if any), we can therefore allow students' minds to 'run wild'. The limitation of course is our own lack of creativity. While it gives us some tools students are still limited by their teachers' limited creativity. On the other hand, teachers might run wild with the idea and create problems that drift too far from the intended topic. For example it is not hard to get from plane geometry to constructing Darboux functions by asking a series of 'what if not' questions.

David Hewitt Video Reflection

David Hewitt used a method of instruction that involved 'enacting', which is to actively use movement and gestures to get a point across. He used tapping on the board and counting the taps to teach counting, drew a number line and the concept of 'jumps' from point to point on the number line to simulate addition and subtraction. While I was quick to judge that this method of instruction trivializes the content and doesn't quite encapsulate the deep foundational roots of the mathematics being taught, I quickly realized the concepts he embeds in his pedagogy. For example, the 'jumps' along the number line quickly makes sense of the concept of a negative number. A negative number cannot be described as a quantity; one cannot 'show' a negative number like one can with a positive number (you can't 'show' -2 apples for example). In particular, the concept of a negative number is that of an OPERATION. One cannot have a negative number without TAKING AWAY something and the action of taking away is the key. This pedagogy would make further explorations in extending 'addition and subtraction' to more abstract abelian binary operations in ring theory much more intuitive. The one critique of his pedagogy is that while he was very engaging he didn't really pose 'real' problems, which one can hope to find in the realm of applied mathematics (like the math behind earthquakes, celestial systems, etc.)

Two Column Problem Solving

Problem (Liouville): Let n > 0 be a positive integer. For each positive divisor r of n count the number of divisors of r. Suppose that the number of divisors of r is s_r. Let S be the square of the sum of the s_r's. Let T be the sum of the cubes of the s_r's. What is the relationship between S and T?

First Column:

Case 1: n = p, p a prime.

In this case the only divisors of n are 1 and p. 1 has 1 positive divisor and p has 2. Hence s_1 = 1 and s_p = 2. S = (1 + 2)^2 = 9. The sum of their cubes is 1^3 + 2^3 = 9.

Case 2: n = pq, p and q distinct primes.

In this case the divisors of n are 1, p, q, and pq. s_1 = 1, s_p = s_q = 2, and s_{pq} = 4. Thus S = (1 + 2 + 2 + 4)^2 = (1+2 +2 (1+2))^2 = (1+2)^2 * (1+2)^2= 81. T = 1^3 + 2^3 + 2^3 + 4^3 = (1^2 + 2^3 + 2^3(1 + 2^3))= (1^3 + 2^3)(1^3 + 2^3) = 9*9 = 81.

Case 3: n = p^m, p a prime and m > 1 is a positive integer.

In this case the divisors of n are 1, p, p^2, ... p^m. s_1 = 1, s_p = 2, s_{p^2} = 3, and in general s_{p^j} = j+1. Thus S = (1 + 2 + 3 + ... + (m+1))^2 = [(m+1)(m+2)/2]^2. T = (1^3 + 2^3 + ... + (m+1)^3) = [(m+1)(m+2)/2]^2 by the summation of cubes formula.

Case 4: n = (p^a)(q^b) where p,q are distinct primes and a, b > 1 are positive integers.

In this case the divisors of n are of the form (p^i)(q^j), where 0 =< i =< a, 0 =< j =< b. After reorganizing we can write S = [(1 + 2 + ... + (a+1)) + 2(1 + 2 + ... +(a+1)) + ... + (b+1)(1 + 2 + ... + (a+1))]^2 = [(1 + 2 + ... + (a+1))^2][1 + 2 + ... + (b+1)]^2 = [(a+1)(a+2)/2]^2 * [(b+1)(b+2)/2]^2 . Similarly, after reorganizing we find T = (1^3 + 2^3 + ... + (a+1)^3)(1 + 2 + ... + (b+1)^3) which by the previous case is equal to S.

The general case follows by induction on k, where n = (p_1)^(a_1) * (p_2)^(a_2) * ... * (p_k)^(a_k) where the p_i's are distinct primes and a_i's are positive integers.

Second Column:

The problem looks very interesting because the wording problem strongly suggests finding an unexpected equality. As the problem is all about the NUMBER of divisors and nothing about the size of the divisors it strongly suggests that the SIZE of the prime factors in the factorization of n doesn't matter, it is all about the NUMBER of divisors. It then make sense to first explore the case when n is a prime.

After examining case 1 our suspicions are confirmed when we find that regardless of which prime p is, the value of S (as defined on the first column) remains the same. We then boldly explore perhaps more difficult cases, which leads us to case 2. In this simple case we found a method to 'reorganize' the terms involved in the sum to obtain S and T respectively that we hope may generalize.

We find that case 2 is not a sufficient 'work horse' to solve the general result. We need something to deal with the case when n has arbitrarily large prime powers as factors. The simplest such case of course is when n = p^m, a power of a prime. This is what we choose as our third case.

Eureka! Third case gave us something incredibly illuminating. It seems that Liouville is attempting to find a more general case of of the sum of cubes formula, which is exactly what turns out to be the case when n is the power of a prime. This is the 'work horse' that we are looking for. But just to be safe, we should see if we can combine the insights obtained in case 2 and 3 together by consider when n is the product of two distinct prime powers.

And we see that the factoring and re-organizing tactic employed in the second case can be applied fruitfully in conjunction with the third case to provide a satisfying and complete solution to case 4. Thus, the rest of the problem follows easily by induction.

This problem is reminiscent of the typical train of thought of research mathematicians of that era, an emphasis on searching for new and surprising equalities. This result is a relatively deep and interesting extension on the fact that the number of factors of a number only depends on how many prime factors it has and to what powers these primes are raised, and not on the value of the primes themselves.

Short Practicum Story

One particularly interesting story during my short practicum is when I observed a social studies class. It was a grade 10 enriched social studies class, otherwise known as pre-IB. A group of three students was presenting on an international political issue, that of the nuclear program development in Iran. The students discussed the issue from the perspective of multiple international stakeholders, including the USA, NATO, and the UN. It was interesting to see the nascent stages of their insight into international relations and history. However, I found that their presentation to be severely lacking in that they missed multiple key players in the discussion (most notably the EU). I also observed the interaction of the teacher with the students and his effectiveness in guiding critical thinking. Based on these topics however I find it interesting to explore the possibility of integrating topics that would otherwise be considered a humanities topic in a mathematical context (for example, the mathematical fairness of the selection process for the UN security council).

Micro Teaching Reflection

The preparation for this assignment was somewhat rushed. The bridging activity was pretty good because it connected seamlessly with the weekend that just passed and the students were able to identify with the story. The lack of rehearsal was evident in the presentation was the group argued a bit with each other during transitions and other parts of the presentation. The classroom set up wasn't as optimized since some people couldn't see parts of the board while the proof was being presented. The review exercise was too 'academic' to really be a fun game and the class wasn't as engaged as we thought. But overall the presentation was still good as most of the mathematical ideas were presented in a transparent fashion that we thought was easy to understand.

10 Questions to the Authors of 'The Art of Problem Posing'

1. Is it appropriate to abstract the statement of a problem or deliberately leave it in some 'simple' form? (For example, "show that it is impossible to write 2009 as the sum of two even numbers" versus "prove that an odd number cannot be expressed as the sum of two even numbers")

2. What do you think of "prove or disprove" type questions?

3. How does one first introduce problems that do not involve numerical answers? (For example, proving that a power series converges uniformly within its radius of convergence)

4. Should problem solving be taught as a technique or as a vehicle to reach other results/techniques?

5. What expectation should one hold regarding how many students are able to solve a given problem?

6. Is it wise to pose problems that go beyond the immediate boundaries of 'what has already been taught' to stimulate interest and to pique desire to explore?

7. How should one interpret the term 'elementary mathematics' when it comes to posing problems to secondary students?

8. Is there any difference in creating a problem designed for math competitions versus for daily use in a classroom?

9. How do we pose problems in such a way so that students do not become 'dependent' on context and heuristics?

10. How do you design a question to urge students to ask 'why'?

Future Students View Points

Grigori Tao

Among the numerous people who have contributed to my success at the International Mathematical Olympiad the first I wish to recognize is Mr. Stanley Xiao, my math teacher from grade 10 to 12.

Mr. Xiao was a very unique mathematics teacher. First, unlike the majority of teachers out there Mr. Xiao is a bonafide ace mathematician whose knowledge and skill level parallels many professional mathematicians. The fact that he chose to be a math teacher attests to his passion and desire to be here. He is a teacher because he wants to be, not because he has to be. In turn his passion for mathematics and his emphasis on excellence affected all in his class. Second, he is the most passionate fan of mathematics competitions. Frequently in class he would talk about various national and international level math competitions, anecdotes about past contestants, how to get there, and how he is willing to train any and all wishing to go down that path. He treats math competitions as seriously as most people do professional sports and that made the mathletes feel very good about themselves. Finally, he constantly challenged the class with difficult problem sets that were for grades that constantly motivated us to think and act like mathematicians. Without him, I would not have had the skill, inspiration, and motivation needed to do well enough on the Canadian Open Math Challenge and subsequently the Canadian Mathematical Olympiad and the USA Mathematical Olympiad to qualify for the International Mathematical Olympiad, much less returning home as a gold medalist.

Indeed, my experiences are not unique. In his 10 years of teaching 4 students that he has taught has returned home as IMO gold medalists, more than the combined total from BC from 1980 (Canada's first year at the IMO) to 2010. I have recommended that Mr. Xiao be part of the IMO training staff in the future to continue his excellent work.

For his inspiration and countless hours of helping us learn to problem solve and challenging us to be the best we can be, I thank him with the utmost sincerity.

Nora Lau

In retrospect of my high school education, it is reminescent of the worst kind of mentality of most major universities: that is, the idea that all chemistry/physics/English/biology/geography/history/...etc. students are to be trained in such a way that they ought to be future researchers in the subject area. The one teacher I had that most underline this mentality is one Mr. Stanley Xiao. In many ways, Mr. Xiao is perhaps objectively an excellent teacher. Indeed, under his tutelage four people (Grigori Tao, Ronald Barton, Jonathan Reid, Bruno Zhai) have went to the International Mathematical Olympiad and returned as medalists. Many more scored extraordinarily high on national and international level math competitions. However, despite his glorious record, his teaching style has isolated at least as many people, including myself. I have never proclaimed to be very strong at math, nor am I very interested in having much to do with mathematics past the immediate need of my field of study (commerce). Mr. Xiao didn't believe in accommodating people of my desires and continuously bombarded us with difficult assignments and tests, forcing us to spend an exorbitant amount of time on our math courses. In the end, much of what we learned was unnecessary. This of course was not restricted to just Mr. Xiao but presents a pandemic among all of academia.

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.