Now that my senior project is coming to an end, I have been looking back and reflecting on my mathematical journey, both over the past month and in my secondary education in general. One thing I realized about my experiences was that when I was challenged I did my best and learned the most. This sounds simple, but there's a lot to be gained from this.
Just like what Polya said in his book about the "right" difficulty of problems for a young student, there is a certain range of problems I find ideal. For the purposes of this post I will divide all math problems into three categories.
1) Trivial/Easy
2) Challenging, but solvable
3) Impossible
Guess which one I am going to advocate ;)
Seriously, the majority of math problems I have done in my life have been easy. You apply a formula/theorem, do some rote calculations, and you win. After doing a certain number of near identical problems, there is nothing more to be gained from these types of problems. Fortunately, math curriculum in a school moves along fast enough that you do not spend an entire year doing the Quadratic Formula or applying the Mean Value Theorem. Especially in a topic like calculus, many of the problems are of this nature simply because the AP curriculum focuses on computational calculus much more than the theory behind it.
The other extreme, if you will, of a problem is an impossible problem. While there are few truly impossible problems, there are many problems that are impossible for a given student given his/her background and knowledge. There is a reason why 9th graders aren't taught Calculus, forget something like Differential Equations or Topology. They simply are not prepared for the material. A key aspect of an impossible problem is that the student does not have the facilities to solve it. Even my beloved, mathematically inclined best friend Daniel Heins looks at certain problems and knows he cannot solve them. In Polya's book, he argues that it is the responsibility of teachers to avoid presenting students with problems they have no legitimate chance of succeeding on. He also believes that teachers are responsible for introducing the ideal type of problem, a challenging one!
Describing what exactly a challenging problem consists of is no cake walk. However, there are some key themes.
1) The problem should take more than a minute to solve.
2) The problem should blend already known things with unknowns.
3) The problem's solution should contain some insights into similar problems in general.
4) The problem's solution may require the student to apply previous knowledge in a new way/method.
5) The problem should challenge the student enough to foster ingenuity and creativity without rendering the student helpless in the situation.
If the student has never worked with modular before, then asking what 17 mod 3 equals is pointless. However, after a few mod problems have been reviewed, a challenging yet achievable problem would be converting something in mod 3 to mod 7 or something.
I wish I had openly tackled more of the challenging problems I was given in high school. This past year, in Mr. Markey's calculus class, the last two/three problems tended to be open ended and somewhat challenging. Being the immature student that I am, I toyed with these problems for a minute or two before giving up, as I had already completed the rest of the problem set.
Only through challenging yourself can you find the extent of your knowledge and continue to improve. There's a reason the Lakers made it to the finals, and it wasn't because they played high school basketball teams. They played a lot of teams that challenged them and forced them to improve. Why not challenge yourself in math? Find these challenging problems, try them, fail at them, and try them again. Then talk with a teacher/friend and find out the solution. You often learn just as much by struggling through a problem and not solving it as you learned from solving it, as you can find out where your mind failed you and whether or not your approach was on track or not.
The rest of your life is full of challenges, some of which are too easy, some of which are too hard, but most of which are difficult but achievable. Why should your mathematics experience be any different?
Wednesday, June 9, 2010
Sunday, June 6, 2010
Lesson Five: Rely less on your textbook
One of the most important things I have realized during my math blog is how frequently my math book leads me astray and into dangerous territory. What could Andrew possibly be talking about? I thought math books were perfect!
Example problems!
Initially I thought example problems were key in helping me understand the material. Looking back, I notice how much they have hindered me. In the math book I have been working through, they introduce new topics with a paragraph or two. Then, you are asked to solve a series of basic problems covering the material. These problems are carefully chosen, as each one teaches the student a bit more about the topic being covered. Then more advanced material relating to the topic is introduced through a series of new problems. Instead of the book solving "key" problems for me, I must first try and struggle through them.
On a personal note, I have found it is much more satisfying to understand the implications of a certain theorem or concept on my own than having my book explain it to me. Using this method, the mathematics student is challenged every step of the way, forced to think through unknowns and apply new ideas to problems.
I found this to be much different than my own math homework, where I get to do a few problems practicing X formula/method, which is easily found in the example problems. I got really good at glancing at example problems and being able to think only in terms of these 5-6 key problems per chapter.
An example of the effects of narrowing a student's mindset:
On my chapter five calculus test (optimization, critical points, etc.) there was an optimization problem relating to minimizing the cost to make a cube. This was a standard optimization problem. However Mr. Markey, the sly dog that he is, added cost as an X factor, something the book hadn't done. We were fine at minimizing the meters of fence needed, but once price was introduced my class struggled. Few people got this problem right, even though all that was necessary was multiplying the various lengths (bottom, sides, etc.) by their respective prices before optimizing. This, however, was a twist on the example problem. Sadly, because we had been conditioned to solve problems of an almost identical nature to the example problem in the textbook, this trivial problem became a great challenge.
While just one example, I think it highlights the dangers of relying too much on a book to provide typical problems. I recently flipped through my calculus book (I only do this a couple times a week =P ) and tried to find problems that were more like the problem solving problems I have been doing recently. It turns out the best problems in the entire section were at the end, as these problems were not cut and dry and instead combined ideas from the section with previous sections/chapters. Furhtermore, these problems were not under the header of "Use X theorem" or "Use Y method". This forces students to think critically and rely on all four steps of problem solving more.
Sorry for the rant-like nature of this post, it was just something that I noticed that really frustrated me.
Cliffnotes: Find problems that are related to the topic you are studying/learning about, but yet where the solution is not easily accessible. If you know how to solve the problem within 5-10 seconds, chances are it's too easy for you! Challenge yourself and you will do better in math, I promise!
Example problems!
Initially I thought example problems were key in helping me understand the material. Looking back, I notice how much they have hindered me. In the math book I have been working through, they introduce new topics with a paragraph or two. Then, you are asked to solve a series of basic problems covering the material. These problems are carefully chosen, as each one teaches the student a bit more about the topic being covered. Then more advanced material relating to the topic is introduced through a series of new problems. Instead of the book solving "key" problems for me, I must first try and struggle through them.
On a personal note, I have found it is much more satisfying to understand the implications of a certain theorem or concept on my own than having my book explain it to me. Using this method, the mathematics student is challenged every step of the way, forced to think through unknowns and apply new ideas to problems.
I found this to be much different than my own math homework, where I get to do a few problems practicing X formula/method, which is easily found in the example problems. I got really good at glancing at example problems and being able to think only in terms of these 5-6 key problems per chapter.
An example of the effects of narrowing a student's mindset:
On my chapter five calculus test (optimization, critical points, etc.) there was an optimization problem relating to minimizing the cost to make a cube. This was a standard optimization problem. However Mr. Markey, the sly dog that he is, added cost as an X factor, something the book hadn't done. We were fine at minimizing the meters of fence needed, but once price was introduced my class struggled. Few people got this problem right, even though all that was necessary was multiplying the various lengths (bottom, sides, etc.) by their respective prices before optimizing. This, however, was a twist on the example problem. Sadly, because we had been conditioned to solve problems of an almost identical nature to the example problem in the textbook, this trivial problem became a great challenge.
While just one example, I think it highlights the dangers of relying too much on a book to provide typical problems. I recently flipped through my calculus book (I only do this a couple times a week =P ) and tried to find problems that were more like the problem solving problems I have been doing recently. It turns out the best problems in the entire section were at the end, as these problems were not cut and dry and instead combined ideas from the section with previous sections/chapters. Furhtermore, these problems were not under the header of "Use X theorem" or "Use Y method". This forces students to think critically and rely on all four steps of problem solving more.
Sorry for the rant-like nature of this post, it was just something that I noticed that really frustrated me.
Cliffnotes: Find problems that are related to the topic you are studying/learning about, but yet where the solution is not easily accessible. If you know how to solve the problem within 5-10 seconds, chances are it's too easy for you! Challenge yourself and you will do better in math, I promise!
Thursday, June 3, 2010
Just for Fun: Proving the formula for diagonals of an N-gon
Proof: The number of diagonals of a polygon with N vertices (N-gon) can be expressed by Diagonals=[N(N-3)]/2.
Instead of looking at base cases and taking an inductive approach, let's just pick a vertice.
1) In a polygon with N vertices, we can pick one of any N vertices.
2) From any vertice, diagonals can be drawn to all of the vertices but 3 (or N-3 vertices).
3) Multiplying N(N-3) covers the total number of diagonals from our randomly selected vertice, as well as if we had picked any of the other N-1 vertices to start from.
4) N(N-3), however, is not correct as that counts the diagonal AB and the diagonal BA as different diagonals.
5) Given this overlap, we simply divide N(N-3) by 2.
6) Total # of Diagonals=D=[N(N-3)]/2. QED.
Hope you enjoyed this as much as I did!
Instead of looking at base cases and taking an inductive approach, let's just pick a vertice.
1) In a polygon with N vertices, we can pick one of any N vertices.
2) From any vertice, diagonals can be drawn to all of the vertices but 3 (or N-3 vertices).
3) Multiplying N(N-3) covers the total number of diagonals from our randomly selected vertice, as well as if we had picked any of the other N-1 vertices to start from.
4) N(N-3), however, is not correct as that counts the diagonal AB and the diagonal BA as different diagonals.
5) Given this overlap, we simply divide N(N-3) by 2.
6) Total # of Diagonals=D=[N(N-3)]/2. QED.
Hope you enjoyed this as much as I did!
Lesson Four: The Power of Diagramming and other Visual Aides
I was looking over some basic combinatorics problems and stumbled upon some very interesting solutions to a few problems. The problems below are not meant to be particularly challenging, but instead are meant to demonstrate the power of diagramming.
Here's the first problem:
How many ways are there to put eight rooks on a chessboard so that they do not attack each other?
For those of you not familiar with chess, a chessboard is an 8x8 grid, and rooks attack each other by sharing a row or column with another rook.
To begin with, the rook can be placed in any of eight columns. The second rook can be placed in any of seven columns. This continues on until the rook is forced to be placed in the final column. The answer is 8! (or 8x7x6x5x4x3x2x1). Factorial notation is much simpler to use and occurs frequently in combinatorics problems...so I will use it from now on.
Easy enough, right?
How about this problem:
There are N booys and N girls in a dance class. How many ways are there to arrange them in pairs for a dance?
**** Try it! ****
This problem, as it turns out, is quite similar to the problem above. In fact, the same image of a chess board can be used to help us solve this problem.
Instead of chess pieces, we can think of each boy as a numbered row (1 through 8) and each girl as a column (A through H). See what I'm getting at?
Let's say that each couple placed a dot where their respective column and row met. Assuming a monogamous dance, each row and column can only have one such dot because each person is dancing with EXACTLY one other person. Then, this problem becomes exactly like the rook problem! So...for N girls and N boys we have N! possible pairings.
Can you think of other settings for this type of problem?
I will post a couple in the comments section that I thought of :)
Another simple combinatorics problem that amused me was the following:
There are 18 ping pong players at a ping pong tournament. The tournament is a round robin (where every player plays every other player exactly once). How many matches are played?
Initially, I thought about adding each player's matches in subsequent order. The first player plays 17 unique matches, the second player plays 17 matches, but he plays one match with the first player, making 16 unique matches. This pattern continues until you reach the last player who plays no unique matches.
Instead...the book hinted at representing all the players on an N-gon, or a polygon with N vertices. Using this method, we choose any vertice for one of our 18 players. We then know our one player plays 17 other matches. So we would have 18*17 for each player's 17 other diagonals or matches. This number, however, is incorrect, as it counts the match A plays with B and B plays with A as unique. Therefore, we divide by two. 18*17/2, or 153 total matches.
This blog post provides an excellet segue into the importance of generality, which will be the topic of my next blog post.
Here's the first problem:
How many ways are there to put eight rooks on a chessboard so that they do not attack each other?
For those of you not familiar with chess, a chessboard is an 8x8 grid, and rooks attack each other by sharing a row or column with another rook.
To begin with, the rook can be placed in any of eight columns. The second rook can be placed in any of seven columns. This continues on until the rook is forced to be placed in the final column. The answer is 8! (or 8x7x6x5x4x3x2x1). Factorial notation is much simpler to use and occurs frequently in combinatorics problems...so I will use it from now on.
Easy enough, right?
How about this problem:
There are N booys and N girls in a dance class. How many ways are there to arrange them in pairs for a dance?
**** Try it! ****
This problem, as it turns out, is quite similar to the problem above. In fact, the same image of a chess board can be used to help us solve this problem.
Instead of chess pieces, we can think of each boy as a numbered row (1 through 8) and each girl as a column (A through H). See what I'm getting at?
Let's say that each couple placed a dot where their respective column and row met. Assuming a monogamous dance, each row and column can only have one such dot because each person is dancing with EXACTLY one other person. Then, this problem becomes exactly like the rook problem! So...for N girls and N boys we have N! possible pairings.
Can you think of other settings for this type of problem?
I will post a couple in the comments section that I thought of :)
Another simple combinatorics problem that amused me was the following:
There are 18 ping pong players at a ping pong tournament. The tournament is a round robin (where every player plays every other player exactly once). How many matches are played?
Initially, I thought about adding each player's matches in subsequent order. The first player plays 17 unique matches, the second player plays 17 matches, but he plays one match with the first player, making 16 unique matches. This pattern continues until you reach the last player who plays no unique matches.
Instead...the book hinted at representing all the players on an N-gon, or a polygon with N vertices. Using this method, we choose any vertice for one of our 18 players. We then know our one player plays 17 other matches. So we would have 18*17 for each player's 17 other diagonals or matches. This number, however, is incorrect, as it counts the match A plays with B and B plays with A as unique. Therefore, we divide by two. 18*17/2, or 153 total matches.
This blog post provides an excellet segue into the importance of generality, which will be the topic of my next blog post.
Wednesday, June 2, 2010
Just for Fun: Seven Bridges of Konigsberg Problem & an Introduction to Graph Theory
Graph theory, simply put, is unlike anything I have ever learned in the classroom. The basis of graph theory is using graphs to model relations between various points/objects/nodes. Here is an example of a graph:
The coolest part about the graph above is that the numbers 1,2,3,4,5, and 6 can represent pretty much anything. The numbers can represent houses and the sidewalks linking them, they can represent cities and the highways ocnnecting them, they can represent people and the relationships between them.
Each of the numbers, or dots, is called a vertice or node. The lines connecting the vertices are called edges. Now we will examine the problem that inspired the field of graph theory: The Seven Bridges of Konigsberg.
This is a rough drawing of the seven bridges. The question that tourists always had was whether or not they could cross each bridge exactly once while starting and finishing in the same place. To solve this problem, Euler drew a simplified graph of only vertices and edges. He then ended up with a graph like this.
Instead of each vertice representing a person, each vertices represents a landmass and each edge represents a bridge connecting two landmasses. If you count the number of bridges (edges) going to each vertice, you will notice that they all have odd numbers. 5, 3, 3, and 3. The problem of starting and ending at the same place and crossing each edge only once is referred to as a Eulerian circuit problem. A Eulerian circuit only exists if the graph is connected (meaning no isolated vertices), and all the vertices are of an even degree. The degree of vertices in this problem are all odd, making this a non Eulerian circuit, or an impossible problem for the tourist.
While this in itself is an interesting problem, there are ways to "Eulerize" a graph to make an Eulerian circuit. I will leave this challenge up to you. How can you make this graph an Eulerian circuit given the definition of a Eulerian circuit? I will post a hint in the comments.
Other interesting graph theory applications/problems include:
The Chinese Postman Problem-(http://en.wikipedia.org/wiki/Route_inspection_problem)
Nearest Neighbor Algorith, used in the "Which way do I travel to college" posters found in math classrooms at Sidwell-(http://en.wikipedia.org/wiki/Nearest_neighbour_algorithm)
The coolest part about the graph above is that the numbers 1,2,3,4,5, and 6 can represent pretty much anything. The numbers can represent houses and the sidewalks linking them, they can represent cities and the highways ocnnecting them, they can represent people and the relationships between them.
Each of the numbers, or dots, is called a vertice or node. The lines connecting the vertices are called edges. Now we will examine the problem that inspired the field of graph theory: The Seven Bridges of Konigsberg.
This is a rough drawing of the seven bridges. The question that tourists always had was whether or not they could cross each bridge exactly once while starting and finishing in the same place. To solve this problem, Euler drew a simplified graph of only vertices and edges. He then ended up with a graph like this.
Instead of each vertice representing a person, each vertices represents a landmass and each edge represents a bridge connecting two landmasses. If you count the number of bridges (edges) going to each vertice, you will notice that they all have odd numbers. 5, 3, 3, and 3. The problem of starting and ending at the same place and crossing each edge only once is referred to as a Eulerian circuit problem. A Eulerian circuit only exists if the graph is connected (meaning no isolated vertices), and all the vertices are of an even degree. The degree of vertices in this problem are all odd, making this a non Eulerian circuit, or an impossible problem for the tourist.
While this in itself is an interesting problem, there are ways to "Eulerize" a graph to make an Eulerian circuit. I will leave this challenge up to you. How can you make this graph an Eulerian circuit given the definition of a Eulerian circuit? I will post a hint in the comments.
Other interesting graph theory applications/problems include:
The Chinese Postman Problem-(http://en.wikipedia.org/wiki/Route_inspection_problem)
Nearest Neighbor Algorith, used in the "Which way do I travel to college" posters found in math classrooms at Sidwell-(http://en.wikipedia.org/wiki/Nearest_neighbour_algorithm)
Monday, May 24, 2010
Lesson Three: Thinking outside the box
Last week I met with Mr. Markey, Rachel Gray, and Thomas Esch to discuss some problem solving. After outlining the four steps of problem solving, Mr. Markey introduced a neat way of looking at a basic problem.
Here's the problem. As you solve it, think as many different ways of solving it as possible.
Problem One: There are 64 teams which take part in March Madness each year. How many games are played before a winner is crowned NCAA champion?
***Think about different ways of solving it before reading the answer below***
Mr. Markey presented this alternate way of solving the problem. Instead of trying to calculate each game played using powers of two or something like that, think about the format of the tournament. Each team which does not win the entire tournament loses exactly one game before being kicked out of the tournament. Given that only one team goes undefeated throughout the entire tournament. This implies that 63 of the 64 teams lose exactly one game, meaning that exactly 63 games were played.
While this problems isn't "SUPER HARD", I thought it was a great example of the most elegant solution arising from looking at the problem in a different light than what I initially thought of for solvivg this problem.
Hope you enjoyed it as much as I did.
Here's the problem. As you solve it, think as many different ways of solving it as possible.
Problem One: There are 64 teams which take part in March Madness each year. How many games are played before a winner is crowned NCAA champion?
***Think about different ways of solving it before reading the answer below***
Mr. Markey presented this alternate way of solving the problem. Instead of trying to calculate each game played using powers of two or something like that, think about the format of the tournament. Each team which does not win the entire tournament loses exactly one game before being kicked out of the tournament. Given that only one team goes undefeated throughout the entire tournament. This implies that 63 of the 64 teams lose exactly one game, meaning that exactly 63 games were played.
While this problems isn't "SUPER HARD", I thought it was a great example of the most elegant solution arising from looking at the problem in a different light than what I initially thought of for solvivg this problem.
Hope you enjoyed it as much as I did.
Friday, May 21, 2010
Problem Set One
Apparently, according to a reader of my blog, I have written enough about how to solve problems and have not given actual problems for you to try.
Here are some interesting ones, spanning several "topics" from Mathematical Circles (Russian Experience).
Problem 1: Two children take turns breaking up a rectangular chocolate bar 6 squares wide by 8 squares long. They may break the bar only along the divisions between the squares. If the bar breaks into several pieces, they keep breaking the pieces up until only the individual squares remain. The player who cannot make a break loses the game. Who will win?
Problem 2 (16): Prove that the number n^3 + 2n is divisble by 3 for any natural number n.
Problem 3 (26): How many six-digit numbers have atleast one even digit?
Problem 4 (16): Can one make change of a 25-ruble bill, using in all ten bills each having a value of 1, 3, or 5 rubles?
Problem 5 (5): Find the smallest natural number n such that n! is divisible by 990.
I have intentionally spread these problems across a variety of topics to minimize the mindlessness that many of you experience in a test all on the same specific topic. Try using the four phases. If these are easy, awesome! If not, keep on trying. Either way, I hope the problems above at least exercised your mind a bit.
-Andrew
Here are some interesting ones, spanning several "topics" from Mathematical Circles (Russian Experience).
Problem 1: Two children take turns breaking up a rectangular chocolate bar 6 squares wide by 8 squares long. They may break the bar only along the divisions between the squares. If the bar breaks into several pieces, they keep breaking the pieces up until only the individual squares remain. The player who cannot make a break loses the game. Who will win?
Problem 2 (16): Prove that the number n^3 + 2n is divisble by 3 for any natural number n.
Problem 3 (26): How many six-digit numbers have atleast one even digit?
Problem 4 (16): Can one make change of a 25-ruble bill, using in all ten bills each having a value of 1, 3, or 5 rubles?
Problem 5 (5): Find the smallest natural number n such that n! is divisible by 990.
I have intentionally spread these problems across a variety of topics to minimize the mindlessness that many of you experience in a test all on the same specific topic. Try using the four phases. If these are easy, awesome! If not, keep on trying. Either way, I hope the problems above at least exercised your mind a bit.
-Andrew
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