Recently, I've been working with friends of mine to try to understand various topics in mathematics. Two concepts that have come up flirt closely with the concepts of the infinite but seem to evade some of my understanding regarding such topics. The particular topics here are the notions of algebras of a set and sigma-algebras of a set. An algebra of a set X is a set or list of subsets of X. For any two subsets you find in your algebra, the subset of X that contains all the elements of both must also be in your algebra (which is just a set, remember). Similarly, whenever you pick a subset in your algebra, the complement of that subset with regards to X must also be in the algebra. That means that the set containing all the things in X that are NOT in your selected set (which then makes another subset) is also in your algebra. Finally, for any two subsets in your algebra, the intersection of those subsets (the set containing all the members of BOTH) must also be in the algebra.
So, when you take a set X, like {1, 2, 3, 4} for example, you can write a set of subsets of X by pulling out subsets of X and putting them in your algebra. I'll take the subset {1, 2} of my set X, and put that in my algebra. Because I put that in, the compliment of it must also go in, which is {3, 4}. Now I have two things in my algebra, {1, 2} and {3, 4}. If I take the union of those two, {1, 2, 3, 4}, that ALSO has to go into the algebra! Finally, if I take the intersection of {1, 2} and {3, 4}, which contains nothing (i.e. it's the empty set, {}), that also has to go into the algebra. By making my algebra from the subset {1, 2}, I've been forced to add in more subsets, to get the final structure being { {}, {1, 2}, {3, 4}, and {1, 2, 3, 4} }. All of that had to go in just to make sure that the properties of being an algebra were satisfied. Now, when I take any pair of those and take the union or intersection, I won't leave that list. The same is true if I take any one and take the compliment.
The sigma algebra is not defined by pairwise unions (i.e. taking two subsets from the algebra and taking all the elements from both), but rather from countably many unions. That means that I can grab as many of the subsets in the algebra as I want so long as I pull them out one at a time, in order. Then, I can take all the elements from those selections and the subset containing all of those must be in the algebra. The same is true for intersections.
What I don't get is this: how are those two definitions describing different things? If you can take any number of pairwise combinations and take the union of them one at a time, then you should be able to take the countable combination of the same sets where the order in which you pick them is the order in which you did the pairwise combinations previously. Clearly I'm not understanding some part of this. I'll share the resolution here if I should successfully comprehend what's going on.
Wednesday, September 16, 2009
Thursday, April 16, 2009
Ideals
I find it difficult to view a mathematics education objectively at times. If I try to ponder what should be taught or what the weaknesses of the current curriculum are, I encounter a simple perception-based problem--I enjoyed math at every level. If I'd had my druthers while I was in school, then we'd have seen more sophisticated mathematics courses than what we did have. It is one of my great hopes in becoming a teacher to learn how to transfer some of my success and ardor in the field to my students, and it would be excellent if the way I look at mathematics knowledge worked to do so. I doubt it, but it would be nice. For starters, let me describe this view.
As one advances in mathematics, one continues to be exposed more and more to the theoretical foundations of the field and less and less to the numerical calculations of its use. In some ways, it seems to me that advanced math is more like a language than a set of tools. It's something with which a student can become fluent. It is certainly possible to speak math! When a student learns to do some calculation or to find some important number or point in a math problem, in the language of math it's as though they are speaking rote phrases--the same sorts of phrases one would learn to "get by" in another country. Please, where is the bathroom? How much does this cost? May I have a glass of water? The student can describe some intent but they do not know what the words mean in their statements!
The most successful math teachers I've ever had reserved a significant portion of their exams and quizzes for vocabulary. Starting with Algebra, this is how I was exposed to many years of math. An equation is a relation between two statements. A solution to an equation is the value of the variable that makes the equation true (since clearly in beginning algebra one is not exposed to multiple independent variables). I was armed with the actual language of math, rudimentary as it was at that point, but I knew how to string together sentences and make sense. When a student can bring topics to a point where they make sense, they can understand why they work. Yes, you still have to teach an algebra student to factor equations. The methods of approaching a problem are like the grammar for the language! But, when 'equation', 'root', and 'solution' have proper definitions in the student's mind, I think there will be much less time spent re-teaching the material later because the students will have internalized the concepts more than they do now.
Though I don't teach formally now, I've done my share of tutoring. It is almost always this problem of not knowing what one is actually working with that is the underlying difficulty that students have with applying what they know. Students always seem to be applying methods blindly, without considering whether those methods work with the objects they're talking about. It's worth knowing whether something is essentially a set, or a function, or a logical statement. You cannot grab elements of a function, there's nothing in an equation. The methods and properties that apply to one kind of object will not apply to another. I've seen many people describe the operation of addition or multiplication as being one-to-one, or describe a function as being commutative. It's great that they have enough of a grasp to know that commutativity means things can swap positions, but there's a fundamental misunderstanding if a student is not shown that it's a property that describes some things but not others.
So, I think that one excellent way of approaching mathematics education is to teach it through the language of the field. The language doesn't have to be as sophisticated as it ever gets. No language is learned in such a way. At least you won't be in a position where you expect the student to distill the underlying concept behind a section, which is frequently the definition for a term, without ever actually giving them the chance to understand that definition as it is. They'll start to develop an intuition for how something works. It's wonderful if it gets to the point where they are just almost grasping it, but they can't quite put it to words. Give them the words. They'll keep that grasp that they developed, but might also see further.
This is definitely a more traditional view in the reformed-traditional scale. I'm not saying that one shouldn't pursue getting students to develop their own framework for knowledge, but refining their concepts and providing them with the tools they'll need to actually communicate about the field seems obvious to me. Obvious... that's another term that just gets thrown around. Some take it to mean "I don't know, either!"
As one advances in mathematics, one continues to be exposed more and more to the theoretical foundations of the field and less and less to the numerical calculations of its use. In some ways, it seems to me that advanced math is more like a language than a set of tools. It's something with which a student can become fluent. It is certainly possible to speak math! When a student learns to do some calculation or to find some important number or point in a math problem, in the language of math it's as though they are speaking rote phrases--the same sorts of phrases one would learn to "get by" in another country. Please, where is the bathroom? How much does this cost? May I have a glass of water? The student can describe some intent but they do not know what the words mean in their statements!
The most successful math teachers I've ever had reserved a significant portion of their exams and quizzes for vocabulary. Starting with Algebra, this is how I was exposed to many years of math. An equation is a relation between two statements. A solution to an equation is the value of the variable that makes the equation true (since clearly in beginning algebra one is not exposed to multiple independent variables). I was armed with the actual language of math, rudimentary as it was at that point, but I knew how to string together sentences and make sense. When a student can bring topics to a point where they make sense, they can understand why they work. Yes, you still have to teach an algebra student to factor equations. The methods of approaching a problem are like the grammar for the language! But, when 'equation', 'root', and 'solution' have proper definitions in the student's mind, I think there will be much less time spent re-teaching the material later because the students will have internalized the concepts more than they do now.
Though I don't teach formally now, I've done my share of tutoring. It is almost always this problem of not knowing what one is actually working with that is the underlying difficulty that students have with applying what they know. Students always seem to be applying methods blindly, without considering whether those methods work with the objects they're talking about. It's worth knowing whether something is essentially a set, or a function, or a logical statement. You cannot grab elements of a function, there's nothing in an equation. The methods and properties that apply to one kind of object will not apply to another. I've seen many people describe the operation of addition or multiplication as being one-to-one, or describe a function as being commutative. It's great that they have enough of a grasp to know that commutativity means things can swap positions, but there's a fundamental misunderstanding if a student is not shown that it's a property that describes some things but not others.
So, I think that one excellent way of approaching mathematics education is to teach it through the language of the field. The language doesn't have to be as sophisticated as it ever gets. No language is learned in such a way. At least you won't be in a position where you expect the student to distill the underlying concept behind a section, which is frequently the definition for a term, without ever actually giving them the chance to understand that definition as it is. They'll start to develop an intuition for how something works. It's wonderful if it gets to the point where they are just almost grasping it, but they can't quite put it to words. Give them the words. They'll keep that grasp that they developed, but might also see further.
This is definitely a more traditional view in the reformed-traditional scale. I'm not saying that one shouldn't pursue getting students to develop their own framework for knowledge, but refining their concepts and providing them with the tools they'll need to actually communicate about the field seems obvious to me. Obvious... that's another term that just gets thrown around. Some take it to mean "I don't know, either!"
Monday, February 23, 2009
Starting and Ending with Applications
What can be done to interest a student in a topic in math? I think you show them how the things they do can be related to it. What can be done to make a student interested to continue to do math? You show them how to do something they've always wanted to do by using it. For example, let's take trigonometry and very simple vector applications (which a student would see in any physics class anyway). Trigonometry is a subject that people tend to hate, so I'll choose that one.
A lot of students these days play video games and computer games. One of the classic archetypes of such pastimes is the game from the first-person perspective. In a virtual world, such as the ones created by these games, in order to move right, you press the 'right arrow' button or key. In order to move forward, you press the 'up arrow'. However, some people have gained an edge over other players and seem to be able to move faster due to a trick. See, if they press both at the same time, they can travel diagonally significantly faster than they'd be able to otherwise. Woe to the person trying to flee from such a canny adversary in this game! This works because of vector addition and the rules of right triangles. If you want a game to remove this possible abuse, then how can you change it? The introduction is to use simple equal vectors (meaning the player travels at a 45-degree angle with respect to the direction their avatar faces). Students exposed to trigonometry need a solid background in right triangle geometry, and they can apply that here. It scales up into trig by saying that it doesn't make a whole lot of sense to be able to sidestep as quickly as you can run ahead, and in order to keep the total speed constant, you need to have some function that behaves periodically to monitor it.
The closing topic for trigonometry is to allow the students to try to do something they've wanted to do since you stepped into their lives--pelt you with water balloons. The ballistics of physics can be reduced to a few simple equations and students get a lot of practice solving for specific variables in computing ballistic trajectories. You can approach the problem by having students describe the qualitative behaviors they'd expect from throwing a ball at certain angles to hit a target. Then, you can show them the basic tools they need (they should not be expected to derive ballistic equations unless they've had significant exposure to calculus) in order to use what they've learned to try to hit you with some fun projectile. Some serious set-up is required to do this, since a means of reliably launching a projectile with specific force is sometimes difficult to come by. The trigonometry is in this in that they have to establish a specific angle required for the strength of the launcher, and achieving a simple expression for this requires the use of trig identities.
Physics teachers may complain that you're using up some of their tricks, but motivating a student to do something is a difficult task at times. It's possible, though, if you're willing to get your feet wet...
-Mark
A lot of students these days play video games and computer games. One of the classic archetypes of such pastimes is the game from the first-person perspective. In a virtual world, such as the ones created by these games, in order to move right, you press the 'right arrow' button or key. In order to move forward, you press the 'up arrow'. However, some people have gained an edge over other players and seem to be able to move faster due to a trick. See, if they press both at the same time, they can travel diagonally significantly faster than they'd be able to otherwise. Woe to the person trying to flee from such a canny adversary in this game! This works because of vector addition and the rules of right triangles. If you want a game to remove this possible abuse, then how can you change it? The introduction is to use simple equal vectors (meaning the player travels at a 45-degree angle with respect to the direction their avatar faces). Students exposed to trigonometry need a solid background in right triangle geometry, and they can apply that here. It scales up into trig by saying that it doesn't make a whole lot of sense to be able to sidestep as quickly as you can run ahead, and in order to keep the total speed constant, you need to have some function that behaves periodically to monitor it.
The closing topic for trigonometry is to allow the students to try to do something they've wanted to do since you stepped into their lives--pelt you with water balloons. The ballistics of physics can be reduced to a few simple equations and students get a lot of practice solving for specific variables in computing ballistic trajectories. You can approach the problem by having students describe the qualitative behaviors they'd expect from throwing a ball at certain angles to hit a target. Then, you can show them the basic tools they need (they should not be expected to derive ballistic equations unless they've had significant exposure to calculus) in order to use what they've learned to try to hit you with some fun projectile. Some serious set-up is required to do this, since a means of reliably launching a projectile with specific force is sometimes difficult to come by. The trigonometry is in this in that they have to establish a specific angle required for the strength of the launcher, and achieving a simple expression for this requires the use of trig identities.
Physics teachers may complain that you're using up some of their tricks, but motivating a student to do something is a difficult task at times. It's possible, though, if you're willing to get your feet wet...
-Mark
Friday, February 13, 2009
Neat Factorization
I mentioned in class tonight a crazy method for factoring trinomials that I learned in grade school. I looked here for a reminder of it, since it was such a long time ago.
Here it is, in summary:
For starters, we have a generic trinomial.
Instead of focusing on this equation, we are going to modify it to achieve this form:
Factor this equation, which is made easier by the lack of a large leading coefficient. It will factor into something that looks like this:
Now, plug in ax for each x in that factorization. Then, divide it by a. This is the necessary factorization for the original problem. In other words:
There are some requirements and proofs required to make sure that the a divides evenly from those two factors. But, even if it didn't, it's a successful factorization.
Here it is, in summary:
For starters, we have a generic trinomial.
Monday, February 9, 2009
"Why?"
One of the most fearsome elements in a student's arsenal is the question, "Why?" The power of this interrogation to discover is formidable. Its power to irritate is also great. It is, however, a question that every student should ask themselves whenever something seems wrong. It is not a bad thing for a student to ask, "Why am I learning this?" It's a great opportunity to tell them that it's not pointless. They open up to the possibility of its being used, and the teacher supplies the use.
It is therefore advantageous for a teacher of mathematics to know the utility of each item in a curriculum. It also behooves such a teacher to understand the value of the lessons in mathematics as a whole. While I cannot cover all of the applications of each segment of school math, I can try to address some of the benefits of the classes as a whole.
Mathematics can teach you how to think. This does not mean that everyone will learn to think by doing math, and it certainly doesn't mean that in order to learn to think, you must do math. It is merely one of many subject areas where prolonged exposure develops logic, reason, and memory. Students who would think well using the math model might discover that they have difficulty thinking with other models. It would be a disservice to these students to not permit them the mathematical mind.
Aside from fundamental cognition, mathematics teaches a student to fluidly apply tools to a variety of situations. A fairly elementary problem in ballistics would be nightmarish and impossible without a functional grasp of the concepts of arithmetic, algebra, and (sometimes) calculus. One of the profound mysteries behind mathematics is that it relates to our world at all. Teaching a student to tie the mathematical tools to the real-world concepts that they can sense and recognize is invaluable and I can scarcely believe that anyone gets by without this skill.
Consider a simple problem of paying in change for something. You can count how much you have. The number you get is clearly representative of the value of the money you have. You can physically move the appropriate amount of money for the transaction, counting it out, and then you can again physically count the money you have to get the new number that represents the value. This last step is, of course, simplified dramatically if a student can draw the connection between doling out change and the arithmetic process of subtraction. However, that is not something that should be expected to come naturally to a student. If you really dig down into that, there's no proof that it should be related at all except for experiential coincidence. That's the kind of innovation mathematics can teach you, and you only get more tools as you continue advancing in it.
In fact, once someone is fully armed with school mathematics (arithmetic, geometry, algebra, trigonometry, and a little calculus), one can accomplish staggering feats of computation. The difficult part is actually translating the applicable data from the world into the recognizable forms with which these tools are used. You can take a demand curve from economics and with calculus you can easily discover what price you should charge to get you the maximum amount of money. No one's going to do any better than that! The difficulty is in obtaining that function where you want to apply the tool.
I've always said that mathematics is all about being as lazy as possible while giving up no ground. You cover all your tracks, but take as few steps as possible to get from one place to another. The fact is, school mathematics trivializes a great deal of work that would be required if one could not apply its tools to situations in the real world. This leaves more time for the fun stuff. We all want more time for the fun stuff.
It is therefore advantageous for a teacher of mathematics to know the utility of each item in a curriculum. It also behooves such a teacher to understand the value of the lessons in mathematics as a whole. While I cannot cover all of the applications of each segment of school math, I can try to address some of the benefits of the classes as a whole.
Mathematics can teach you how to think. This does not mean that everyone will learn to think by doing math, and it certainly doesn't mean that in order to learn to think, you must do math. It is merely one of many subject areas where prolonged exposure develops logic, reason, and memory. Students who would think well using the math model might discover that they have difficulty thinking with other models. It would be a disservice to these students to not permit them the mathematical mind.
Aside from fundamental cognition, mathematics teaches a student to fluidly apply tools to a variety of situations. A fairly elementary problem in ballistics would be nightmarish and impossible without a functional grasp of the concepts of arithmetic, algebra, and (sometimes) calculus. One of the profound mysteries behind mathematics is that it relates to our world at all. Teaching a student to tie the mathematical tools to the real-world concepts that they can sense and recognize is invaluable and I can scarcely believe that anyone gets by without this skill.
Consider a simple problem of paying in change for something. You can count how much you have. The number you get is clearly representative of the value of the money you have. You can physically move the appropriate amount of money for the transaction, counting it out, and then you can again physically count the money you have to get the new number that represents the value. This last step is, of course, simplified dramatically if a student can draw the connection between doling out change and the arithmetic process of subtraction. However, that is not something that should be expected to come naturally to a student. If you really dig down into that, there's no proof that it should be related at all except for experiential coincidence. That's the kind of innovation mathematics can teach you, and you only get more tools as you continue advancing in it.
In fact, once someone is fully armed with school mathematics (arithmetic, geometry, algebra, trigonometry, and a little calculus), one can accomplish staggering feats of computation. The difficult part is actually translating the applicable data from the world into the recognizable forms with which these tools are used. You can take a demand curve from economics and with calculus you can easily discover what price you should charge to get you the maximum amount of money. No one's going to do any better than that! The difficulty is in obtaining that function where you want to apply the tool.
I've always said that mathematics is all about being as lazy as possible while giving up no ground. You cover all your tracks, but take as few steps as possible to get from one place to another. The fact is, school mathematics trivializes a great deal of work that would be required if one could not apply its tools to situations in the real world. This leaves more time for the fun stuff. We all want more time for the fun stuff.
Wednesday, January 28, 2009
The Bottom Line
Proof matters. It's the name of the game. I'm a mathematician and I'm going to be a teacher and it's not wrong to get students into that mindset. Proof is justification. Proof is the confidence in knowing that you're right, because you can fully explain your point of view. Proof is covering all your bases. Possibly most importantly, proof is so very often beautiful.
I really enjoy mathematics. I hope to get some others to really enjoy mathematics with me--to bring them to a point where they can appreciate the elegance and austerity of it all. But, one might ask oneself, why should I?
Math is one of the only field of studies that is genuinely sure of the conclusions it draws. It achieves this by being aware of its context. Any student of the subject needs to know that all of mathematics is based on rules. Where those rules apply, mathematics reigns. One only needs to lay the foundations and the citadel appears whole, pristine, and without effort. It is a nearly perfect structure in which every single block rests firmly and precisely on the blocks beneath it. The mortar that holds that structure together is proof. Proof matters.
As I encounter some delightful things in mathematics, I intend to place them here that others may appreciate them. I'm starting this blog under duress. However, maybe I can make something useful and elegant in its own right.
-Mark
I really enjoy mathematics. I hope to get some others to really enjoy mathematics with me--to bring them to a point where they can appreciate the elegance and austerity of it all. But, one might ask oneself, why should I?
Math is one of the only field of studies that is genuinely sure of the conclusions it draws. It achieves this by being aware of its context. Any student of the subject needs to know that all of mathematics is based on rules. Where those rules apply, mathematics reigns. One only needs to lay the foundations and the citadel appears whole, pristine, and without effort. It is a nearly perfect structure in which every single block rests firmly and precisely on the blocks beneath it. The mortar that holds that structure together is proof. Proof matters.
As I encounter some delightful things in mathematics, I intend to place them here that others may appreciate them. I'm starting this blog under duress. However, maybe I can make something useful and elegant in its own right.
-Mark
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