Wednesday, November 25, 2009
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.)
Sunday, November 15, 2009
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.
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.
Friday, November 6, 2009
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).
Friday, October 16, 2009
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.
Sunday, October 4, 2009
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'?
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'?
Friday, October 2, 2009
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.
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.
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