In Math You Have to Remember, In Other Subjects You Can Think About It(maa.org)
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In Math You Have to Remember, In Other Subjects You Can Think About It
http://www.maa.org/devlin/devlin_06_10.html
28 comments
Where can I learn about math notation and how to read math symbols expressions? Many times I find some articule that explain some idea using math symbols. And I can understand nothing. This week I was looking for a method to calculate the distance between two geolocation coordinates. Át. First the article used math symbols to explain and I could understand nothing. I didnt even know how to Google that "enigma". But a few lines below there was the same algorithm, but This time it was written in Javascript. And it was very easy and simple to understand. How to learn "math syntax and grammar"?
Good question. Some resources I found:
http://en.wikipedia.org/wiki/Table_of_mathematical_symbols. It is more of a reference than a tutorial. It also includes only symbols, not other features of notation such as division lines and superscript/subscript.
http://mathoverflow.net/questions/33152/is-there-a-reference.... Someone with a similar, more specific question, looking for a dictionary of what to name variables. Nobody found any such dictionary.
http://www.alcyone.com/max/reference/maths/notation.html. A decently large list of notations. But it doesn’t have, for instance, the notation of calculus, and you must run a web search yourself to learn more about what a listed notation means.
Hopefully in conjunction these resources will help. But it’s too bad that there isn’t anything quite like what you’re looking for; I would be curious to read such a tutorial, too.
http://en.wikipedia.org/wiki/Table_of_mathematical_symbols. It is more of a reference than a tutorial. It also includes only symbols, not other features of notation such as division lines and superscript/subscript.
http://mathoverflow.net/questions/33152/is-there-a-reference.... Someone with a similar, more specific question, looking for a dictionary of what to name variables. Nobody found any such dictionary.
http://www.alcyone.com/max/reference/maths/notation.html. A decently large list of notations. But it doesn’t have, for instance, the notation of calculus, and you must run a web search yourself to learn more about what a listed notation means.
Hopefully in conjunction these resources will help. But it’s too bad that there isn’t anything quite like what you’re looking for; I would be curious to read such a tutorial, too.
Well I was not happy with the plug and chug approach to calculus so I wrote a book that uses a more intuitive geometric approach (at last it is intuitive to me) and put it all online for free at http://www.thegistofcalculus.com but nobody cares to read it so I guess people are actually fine with just memorizing everything.
It is only about 50 pages and explains the meat & potatoes of the math class.
It is only about 50 pages and explains the meat & potatoes of the math class.
This reminds me of one of the math teachers I had, who used a technique I found quite interesting and effective compared to how other teachers taught us about new concepts.
He would rarely come up with some theorem, ask us to solve a few exercises, and go on to the next chapter. When he wanted to introduce a new concept, he'd usually ask us to solve an equation or a more concrete probleme (I can remember something like the fence problem mentioned in the article, which was used to introduce us to finding the maximal output of a 2nd degree expression). After that, we would either find a solution through some struggle — and then try to figure out a more efficient approach — or we'd stumble on a problem by trying to use approaches we'd been taught before — and therefore try to find a better way to represent the problem.
On an amusing sidenote, while we were each working individually, he'd walk around in the class, looking at what we were doing, and crossing out with a red marker pen mistakes we'd made.
He would rarely come up with some theorem, ask us to solve a few exercises, and go on to the next chapter. When he wanted to introduce a new concept, he'd usually ask us to solve an equation or a more concrete probleme (I can remember something like the fence problem mentioned in the article, which was used to introduce us to finding the maximal output of a 2nd degree expression). After that, we would either find a solution through some struggle — and then try to figure out a more efficient approach — or we'd stumble on a problem by trying to use approaches we'd been taught before — and therefore try to find a better way to represent the problem.
On an amusing sidenote, while we were each working individually, he'd walk around in the class, looking at what we were doing, and crossing out with a red marker pen mistakes we'd made.
This captures the problem with mathematical education well enough:
> When Boaler would visit a class being taught in a Railside-like fashion and ask students what they were working on, they would describe the problem and how they were trying to solve it. When she asked the same questions of students being taught the traditional way, they would generally tell them what page of the book they were on. When she asked them, "But what are you actually doing?" they would answer "Oh, I'm doing number 3." [p.98]
The typical math education relies a lot on rote learning (just take a look at the cheatsheets here: http://tutorial.math.lamar.edu/cheat_table.aspx). There are a lot of equations you just have to remember by heart to be able to be productive in solving the typical textbook problems. For a high-school student who does not have any insight on the beauty of mathematics, this is a huge turn off. They leave school with the impression that this is a cold hard subject with perfect proofs that you have to learn by rote, and nothing more. No one talks about why Math is beautiful. The most positive thing about Math I've heard from people is that it is the best subject to get a perfect score. The arts are subjective and there is no perfectly right answer, but Math, if you know how to solve these types of problems without missing a sign or a bracket here or there, you get a 100/100.
The problem, I think, is because we teach the results of hundreds of years of evolution of Maths. For example, most courses on Calculus start teaching it by talking about Limits, and from there moves onto Differentiation. Integration is considered to be the 'advanced' part of Calculus. It was very recently that I discovered Apostle's textbook on Calculus (students of universities who use that text are lucky) where he treats Integration first - because that is the right historical order in which Calculus evolved.
I could appreciate it a lot more when I understood what kind of problems were Newton and Leibniz trying to solve when they came up with the formalized notion of Calculus. Calculus was described by them using the concept of 'infinitesimals', not through Limits. Limits was a clever abstraction that was evolved later to better explain Calculus and keep it consistent. But when we start teaching students Calculus with Limits, show them the perfect way where Limits can be used to find the differentials of trigonometric functions, they do not know this background. For them, there is no moment of 'awe'. They are not even shown a glimpse of the amazing intellectual pursuit that was behind this fantastic subject. All you see are a bunch of equations, some proofs that are mathematically perfect, and you just learn them by rote.
The typical Math education needs to focus more on the evolution of the subject, the pains faced by mathematicians (or physicists!) to which they came up with these solutions. The logical gaps in new ideas and how they were filled later. Let the students understand that this is not a 'perfect' subject. There were human beings who faced real problems who came up with these solutions. Even better, let them understand that some of the things they learn was the result an intellectual pastime for these mathematicians. It was imperfect, and there was joy when mathematicians brought it closer to perfection.
With the advent of computers, most of the evaluation criteria used in High School Math is becoming redundant. Moving from one step to another without making careless mistakes is priority number one now. If we reduce the importance on that manual aspect of the typical Math problem solving, and instead focus on teaching the more interesting, insightful things about Math, the students will go away with a totally different idea about Maths. Like programming, it becomes a universe of abstractions where your curiosity drives you to learn more.
> When Boaler would visit a class being taught in a Railside-like fashion and ask students what they were working on, they would describe the problem and how they were trying to solve it. When she asked the same questions of students being taught the traditional way, they would generally tell them what page of the book they were on. When she asked them, "But what are you actually doing?" they would answer "Oh, I'm doing number 3." [p.98]
The typical math education relies a lot on rote learning (just take a look at the cheatsheets here: http://tutorial.math.lamar.edu/cheat_table.aspx). There are a lot of equations you just have to remember by heart to be able to be productive in solving the typical textbook problems. For a high-school student who does not have any insight on the beauty of mathematics, this is a huge turn off. They leave school with the impression that this is a cold hard subject with perfect proofs that you have to learn by rote, and nothing more. No one talks about why Math is beautiful. The most positive thing about Math I've heard from people is that it is the best subject to get a perfect score. The arts are subjective and there is no perfectly right answer, but Math, if you know how to solve these types of problems without missing a sign or a bracket here or there, you get a 100/100.
The problem, I think, is because we teach the results of hundreds of years of evolution of Maths. For example, most courses on Calculus start teaching it by talking about Limits, and from there moves onto Differentiation. Integration is considered to be the 'advanced' part of Calculus. It was very recently that I discovered Apostle's textbook on Calculus (students of universities who use that text are lucky) where he treats Integration first - because that is the right historical order in which Calculus evolved.
I could appreciate it a lot more when I understood what kind of problems were Newton and Leibniz trying to solve when they came up with the formalized notion of Calculus. Calculus was described by them using the concept of 'infinitesimals', not through Limits. Limits was a clever abstraction that was evolved later to better explain Calculus and keep it consistent. But when we start teaching students Calculus with Limits, show them the perfect way where Limits can be used to find the differentials of trigonometric functions, they do not know this background. For them, there is no moment of 'awe'. They are not even shown a glimpse of the amazing intellectual pursuit that was behind this fantastic subject. All you see are a bunch of equations, some proofs that are mathematically perfect, and you just learn them by rote.
The typical Math education needs to focus more on the evolution of the subject, the pains faced by mathematicians (or physicists!) to which they came up with these solutions. The logical gaps in new ideas and how they were filled later. Let the students understand that this is not a 'perfect' subject. There were human beings who faced real problems who came up with these solutions. Even better, let them understand that some of the things they learn was the result an intellectual pastime for these mathematicians. It was imperfect, and there was joy when mathematicians brought it closer to perfection.
With the advent of computers, most of the evaluation criteria used in High School Math is becoming redundant. Moving from one step to another without making careless mistakes is priority number one now. If we reduce the importance on that manual aspect of the typical Math problem solving, and instead focus on teaching the more interesting, insightful things about Math, the students will go away with a totally different idea about Maths. Like programming, it becomes a universe of abstractions where your curiosity drives you to learn more.
The typical math education relies a lot on rote learning
This has been known and lamented for a century, and it long ago transitioned from being a radical idea to being conventional wisdom. It has been orthodoxy for decades (although recently there has been some reaction against the orthodoxy by a few experimental charter schools.) So, if current math education involves a lot of rote learning, it's not for lack of awareness or lack of attempts to fix the problem. In rebellion against facts and rote learning, teachers have elevated understanding above problem-solving as the highest goal of mathematics education, but it turns out to be impossible to demonstrate understanding, much less engage real-world problems, without knowing a lot of handy facts. That's why the most idealistic, anti-rote-learning teachers still want their kids to learn that sin^2(x) + cos^2(x) = 1. They can't demonstrate the real-world relevance of mathematics or help kids understand abstract concepts unless the kids have some basic problem-solving competence that doesn't involve spending hours searching through the book for fundamental facts.
And yes, they do want kids to get to the point where they can "forget" that sin^2(x) + cos^2(x) = 1 and regenerate that knowledge from visualizing the unit circle and applying the Pythagorean Theorem. That ideal has also been orthodoxy for decades. It just turns out that in practice, deeper understanding emerges from practice, and practice requires competence. Rote learning is a way of bootstrapping competence so competence can be turned into understanding. Believe me, progressive, "kinder and gentler" teachers were the norm in all the schools I've gone to, and they all tried every trick they knew to help us skip straight from ignorance to deep understanding with as little dry, repetitive work as possible, but they could not get around the need to learn many facts and techniques by heart.
By the way, the route through calculus you describe reflects that you are basically the kind of kid that the schools don't worry about. I was the same kind of student. It might seem strange to us, but to most students, the history of a subject and the problems that originally motivated it are the very definition of dry and disconnected. Nobody gives a damn about the awesome intellectual journeys of some dead nerds except kids like you and me who will learn the subject regardless of how it's taught. Differentiation is introduced first precisely because it makes the subject less dry to most students, because it allows them to immediately apply what they learn to simple concrete problems involving rates.
This has been known and lamented for a century, and it long ago transitioned from being a radical idea to being conventional wisdom. It has been orthodoxy for decades (although recently there has been some reaction against the orthodoxy by a few experimental charter schools.) So, if current math education involves a lot of rote learning, it's not for lack of awareness or lack of attempts to fix the problem. In rebellion against facts and rote learning, teachers have elevated understanding above problem-solving as the highest goal of mathematics education, but it turns out to be impossible to demonstrate understanding, much less engage real-world problems, without knowing a lot of handy facts. That's why the most idealistic, anti-rote-learning teachers still want their kids to learn that sin^2(x) + cos^2(x) = 1. They can't demonstrate the real-world relevance of mathematics or help kids understand abstract concepts unless the kids have some basic problem-solving competence that doesn't involve spending hours searching through the book for fundamental facts.
And yes, they do want kids to get to the point where they can "forget" that sin^2(x) + cos^2(x) = 1 and regenerate that knowledge from visualizing the unit circle and applying the Pythagorean Theorem. That ideal has also been orthodoxy for decades. It just turns out that in practice, deeper understanding emerges from practice, and practice requires competence. Rote learning is a way of bootstrapping competence so competence can be turned into understanding. Believe me, progressive, "kinder and gentler" teachers were the norm in all the schools I've gone to, and they all tried every trick they knew to help us skip straight from ignorance to deep understanding with as little dry, repetitive work as possible, but they could not get around the need to learn many facts and techniques by heart.
By the way, the route through calculus you describe reflects that you are basically the kind of kid that the schools don't worry about. I was the same kind of student. It might seem strange to us, but to most students, the history of a subject and the problems that originally motivated it are the very definition of dry and disconnected. Nobody gives a damn about the awesome intellectual journeys of some dead nerds except kids like you and me who will learn the subject regardless of how it's taught. Differentiation is introduced first precisely because it makes the subject less dry to most students, because it allows them to immediately apply what they learn to simple concrete problems involving rates.
I agree mostly with your comment (check out my sibling one).
>but it turns out to be impossible to demonstrate understanding, much less engage real-world problems, without knowing a lot of handy facts. That's why the most idealistic, anti-rote-learning teachers still want their kids to learn that sin^2(x) + cos^2(x) = 1. They can't demonstrate the real-world relevance of mathematics or help kids understand abstract concepts unless the kids have some basic problem-solving competence that doesn't involve spending hours searching through the book for fundamental facts.
Might I suggest that they take the top-down approach instead of the bottom-up one? If you have a clearly defined real world problem (no matter how simplistic), it's easy to see what information is missing and needed, and so it's then clear why a certain proof or relation is required. Then it's more reasonable to delve into the theoretical and abstract, when it is required by the real world.
Then you can make the real world problems harder to delve deeper into the theoreticals. You're more grounded in this way.
When you take the opposite approach and teach sin^2(x) + cos^2(x) = 1 starting from lines, angles, triangle relations, even your best students say "so fucking what?". In fact, I'm fairly certain that the teachers I had who could keep the whole class' attention and who could teach even the weakest students consistently displayed the top-down approach.
Edit: As an example, I first encountered trigonometry when I was building a rubber-powered trebuchet that I made for a middle school physics class. You could move a crank to lower or heighten the part the rubber was attached to, effectively changing the angle and thus the shooting distance. My dad was showing me how to calculate the angle of the elastic to the ground by knowing the length of the two sides. I instantly "got" why we needed trigonometry and could visualise how changes in the lengths of sides would affect the angles. It seems intuitive now but it wasn't at the time. Of course, I wouldn't solve problems until much later but I feel like a lot of my mates never made that same leap.
>but it turns out to be impossible to demonstrate understanding, much less engage real-world problems, without knowing a lot of handy facts. That's why the most idealistic, anti-rote-learning teachers still want their kids to learn that sin^2(x) + cos^2(x) = 1. They can't demonstrate the real-world relevance of mathematics or help kids understand abstract concepts unless the kids have some basic problem-solving competence that doesn't involve spending hours searching through the book for fundamental facts.
Might I suggest that they take the top-down approach instead of the bottom-up one? If you have a clearly defined real world problem (no matter how simplistic), it's easy to see what information is missing and needed, and so it's then clear why a certain proof or relation is required. Then it's more reasonable to delve into the theoretical and abstract, when it is required by the real world.
Then you can make the real world problems harder to delve deeper into the theoreticals. You're more grounded in this way.
When you take the opposite approach and teach sin^2(x) + cos^2(x) = 1 starting from lines, angles, triangle relations, even your best students say "so fucking what?". In fact, I'm fairly certain that the teachers I had who could keep the whole class' attention and who could teach even the weakest students consistently displayed the top-down approach.
Edit: As an example, I first encountered trigonometry when I was building a rubber-powered trebuchet that I made for a middle school physics class. You could move a crank to lower or heighten the part the rubber was attached to, effectively changing the angle and thus the shooting distance. My dad was showing me how to calculate the angle of the elastic to the ground by knowing the length of the two sides. I instantly "got" why we needed trigonometry and could visualise how changes in the lengths of sides would affect the angles. It seems intuitive now but it wasn't at the time. Of course, I wouldn't solve problems until much later but I feel like a lot of my mates never made that same leap.
Finding some connection between trigonometry and angles of rubber-powered trebuchets is incidental and quite frankly, this approach isn't going to take you very far. One simply can't find real world mappings to atleast 99% of what is properly called math. Even if you'd like to strictly operate within the confines of high school geometry , it is a study of points, lines, planes, angle relations, norms & distances motivated by axioms & theorems that follow by symbol pushing. For example, it so happens that if you push the right symbols, you'll eventually work out that the area of a triangle is related to its sides ( http://en.wikipedia.org/wiki/Herons_formula ). But this is a consequence of your definition of area and triangle and side and so forth. The relationship falls out of those definitions. Its not about constructing a real wooden triangle out of real sticks & then measuring its area and then the lengths of its sticks and saying - hey, I told you so! Those things ie. the real wood and the real sticks,...reality - is just a sideshow. Real world problems are at best a distraction. If you were to focus only on these sort of real-world mappings, you'd end up as an accountant - because that's were 1 would map to $1, 2to $2 and add would mean 1 + 2 = $3 and so forth. That's what corresponds to reality and the real world. That isn't math. Math is where you say 1 + 2 = 0 because the abelian group G3 with three elements {0,1,2} can satisfy the addition rule and have three distinct elements only if 1 + 2 was 0. For if 1 + 2 was 1, then that would mean 2 was equal to zero by cancelation laws. Similarly if 1+2 was 2 then 1 would equal 0. Since you do have 3 elements, it stands to reason that 1+2 must be 0, because it can't be anything else in that system. That's math.
I heartily disagree with your initial point that you can only get so far with real world examples. Heron's formula itself is not even unintuitive, and in fact, the way he proved it, requires a minimal amount of symbol pushing. Maybe my opinion is unpopular, but I believe that anything that is not applicable to the real world and does not form the immediate base of anything that is applicable to the real world has no place in highschool education.
Those of us who do math for math's sake are mathematicians. For the rest of us, math is a tool, no better or worse than any other tools we have in problem solving. There is nothing wrong with benefiting solely from the end conclusions of math, without engaging in it unless we have to. For the majority of people, this suffices and anything more has no benefit, and is promptly forgotten.
So why, then, do we insist that a high school program, tailored to the majority of people, teach people to be proficient at theorizing rather than applying? The majority are accountants ... and doctors, and lawyers, and delivery people, and repairmen, and so on. People can learn about abelian groups in the future, when they know what they want to study. Let's be realistic here. Right now, we're dealing with people not being able to do middle school material 2 years out of college. We're trying to prepare people for what they're in for. For the sake of the public good, abelian groups and math as a whole can wait.
Those of us who do math for math's sake are mathematicians. For the rest of us, math is a tool, no better or worse than any other tools we have in problem solving. There is nothing wrong with benefiting solely from the end conclusions of math, without engaging in it unless we have to. For the majority of people, this suffices and anything more has no benefit, and is promptly forgotten.
So why, then, do we insist that a high school program, tailored to the majority of people, teach people to be proficient at theorizing rather than applying? The majority are accountants ... and doctors, and lawyers, and delivery people, and repairmen, and so on. People can learn about abelian groups in the future, when they know what they want to study. Let's be realistic here. Right now, we're dealing with people not being able to do middle school material 2 years out of college. We're trying to prepare people for what they're in for. For the sake of the public good, abelian groups and math as a whole can wait.
>For the sake of the public good, abelian groups can wait.
Amazing. I can't wait to put that on a plaque. "Abelian groups can wait for the sake of public good." Holy cow! Dude, Niels Abel was 17 when he invented most of the machinery that goes by the name Group Theory. By the age of 19, he had proved quintics don't have a general solution by simply cranking up his machinery. And today we have 17 and 19 year olds who not only don't know what a group is, but don't particularly want to know. Because, like you say, they'd rather be real world accountants & repairmen, not ivory tower mathematicians. Wouldn't you rather have just 1 Abel and a whole generation of pissed off accountants than the other way around ? I would. You know, when we do eventually get the hell out of this planet and conquer other dimensions and populate new worlds, get beyond this ethereal realm so to speak, it would be purely due to the ideas of an Abel or a Gauss or a Riemann. Even an army of real-life accountants wouldn't get you off this planet - they'd be busy calculating the price of the spacecraft with their fancy spreadsheets.
Amazing. I can't wait to put that on a plaque. "Abelian groups can wait for the sake of public good." Holy cow! Dude, Niels Abel was 17 when he invented most of the machinery that goes by the name Group Theory. By the age of 19, he had proved quintics don't have a general solution by simply cranking up his machinery. And today we have 17 and 19 year olds who not only don't know what a group is, but don't particularly want to know. Because, like you say, they'd rather be real world accountants & repairmen, not ivory tower mathematicians. Wouldn't you rather have just 1 Abel and a whole generation of pissed off accountants than the other way around ? I would. You know, when we do eventually get the hell out of this planet and conquer other dimensions and populate new worlds, get beyond this ethereal realm so to speak, it would be purely due to the ideas of an Abel or a Gauss or a Riemann. Even an army of real-life accountants wouldn't get you off this planet - they'd be busy calculating the price of the spacecraft with their fancy spreadsheets.
I guess we have different views of what's important. There are maybe 2-3 Abel's per century. Gauss completed some of his greatest work in his early twenties, but that doesn't mean that everyone else did as well as he did. I'd love for everyone to be Gausses and Abels, but the truth is, they aren't. Neither do they want to be. Neither are they unhappy "pissed off accountants" because they aren't, like you describe. Neither are they idiots because they aren't math-savvy, like you imply.
I'd rather human development and quality of life come before space travel. If there are junior Abels among us, they are going to stand out regardless of whether you teach them subject X in highschool or not. Furthermore, they are going to pursue a specialized education (college) in said subject anyway. I don't think not recognizing geniuses is a problem we have since geniuses tend to be fairly resourceful. Thus, we can focus on improving the standard of education for everyone else. I'm not suggesting a ban on the study of mathematics altogether.
There is no causation link as you imagine. No amount of increased math in highschool is going to breed geniuses like Abels. Increased general skills like logic and problem solving is going to result in higher ability and better education for almost everyone.
I'd rather human development and quality of life come before space travel. If there are junior Abels among us, they are going to stand out regardless of whether you teach them subject X in highschool or not. Furthermore, they are going to pursue a specialized education (college) in said subject anyway. I don't think not recognizing geniuses is a problem we have since geniuses tend to be fairly resourceful. Thus, we can focus on improving the standard of education for everyone else. I'm not suggesting a ban on the study of mathematics altogether.
There is no causation link as you imagine. No amount of increased math in highschool is going to breed geniuses like Abels. Increased general skills like logic and problem solving is going to result in higher ability and better education for almost everyone.
You sound like someone from the Konishi polity as described in Diaspora. I don't agree with you. The brain uses an associative learning algorithm and learning difficulty is proportional to how easily a concept can be integrated with what is already known. And it's an iterative process, so new knowledge can be built on old knowledge that was once itself abstract.
So you can start with concrete problems and generalize from there. No one starts out learning about unitary matrices, hilbert spaces, and natural transformations. By the time they get there they can be taught in terms of more concrete things like rotations in 3 dimensions, euclidean spaces and parametric polymorphism (if they are programmers). That is why math is hard when you first start. It's a highly compressed subject where every word is more an index than it is a full concept. So you have to maintain a cascade or hierarchical graph of intuitions.
So you can start with concrete problems and generalize from there. No one starts out learning about unitary matrices, hilbert spaces, and natural transformations. By the time they get there they can be taught in terms of more concrete things like rotations in 3 dimensions, euclidean spaces and parametric polymorphism (if they are programmers). That is why math is hard when you first start. It's a highly compressed subject where every word is more an index than it is a full concept. So you have to maintain a cascade or hierarchical graph of intuitions.
>you can start with concrete problems and generalize from there
You could. But that isn't the only way. You should really read up on the way math is( was ?) taught in gottingen & eastern europe & romania & russia. Speaking of which, I learnt most of the machinery behind matrices in India in high school, with no reference to where it came from or what its applications were. I could compute the determinant of square matrices, compute adjoints, inverses, multiply two matrices....all of this without knowing wtf a matrix was or what its real life use might be. This is true of most high school students in India, simply because of the way the curriculum is structured. Its only when I took at a graduate level computer graphics course in my late 20s did I see that linear transforms were actually matrices, so you could do rotations and translations of vectors by matrix multiplication. So real life applications don't have to precede abstract concepts - you can make a hell a lot of progress by approaching stuff the other way around.
You could. But that isn't the only way. You should really read up on the way math is( was ?) taught in gottingen & eastern europe & romania & russia. Speaking of which, I learnt most of the machinery behind matrices in India in high school, with no reference to where it came from or what its applications were. I could compute the determinant of square matrices, compute adjoints, inverses, multiply two matrices....all of this without knowing wtf a matrix was or what its real life use might be. This is true of most high school students in India, simply because of the way the curriculum is structured. Its only when I took at a graduate level computer graphics course in my late 20s did I see that linear transforms were actually matrices, so you could do rotations and translations of vectors by matrix multiplication. So real life applications don't have to precede abstract concepts - you can make a hell a lot of progress by approaching stuff the other way around.
> I could compute the determinant of square matrices,
I learned to do this too---and then promptly forgot---several times over until someone finally clued me in to the fact that the determinant is the scale factor of the linear transformation associated with the matrix. Then everything clicked.
It's not just that I couldn't remember how to define/calculate the determinant, I also couldn't apply it usefully in my (proof-based) linear algebra class because I didn't have any intuition for it or any way to connect it to other concepts.
I learned to do this too---and then promptly forgot---several times over until someone finally clued me in to the fact that the determinant is the scale factor of the linear transformation associated with the matrix. Then everything clicked.
It's not just that I couldn't remember how to define/calculate the determinant, I also couldn't apply it usefully in my (proof-based) linear algebra class because I didn't have any intuition for it or any way to connect it to other concepts.
As an example, I first encountered trigonometry when I was building a rubber-powered trebuchet that I made for a middle school physics class....
I have to ask, do you realize how much of an outlier you are and how irrelevant your experience is to teaching normal kids mathematics? I mean... you immediately saw value in mathematically defining the relationship between the angle and the length. Teachers don't worry about kids like you (and me, and most of HN probably) because we'll be fine no matter how the class is taught. Wondering about how to get kids like you or me interested in math is like wondering how to get a cat interested in mice, or a Jersey Shore cast member interested in taking his shirt off. Teachers don't waste a second's thought on kids who will flourish regardless of the classroom environment. They worry about the marginal kids who might succeed if taught well but will flounder if taught poorly. A kid who really says "so fucking what?" to trigonometry is going to say "so fucking what?" to your trebuchet example, too. In the unlikely event he gives a damn about the length of the elastic rubber, he can just crank the attachment up and down and watch how the length changes, so who needs math?
The "best students" have sat through enough math classes that they know the right default assumption is that whatever is taught is probably useful. They get that engineering and science (including social science) depend on math, and math builds on previous math, and the high school curriculum doesn't have room for stuff that never comes up again later (okay, with the exception of geometry class, but even that is mostly useful.) Heck, even the "pretty good" students understand that. Those are the kids who get excited about real-world applications of trig: the kids who already get it and never ask "so fucking what?" Plus there's another group of kids who will never seriously ask "so fucking what," the ones who care about grades and college applications. They already know how they're going to use math: they're going to use it to get good grades in math class.
When students in trigonometry class really do say "so fucking what?" and "When are we ever going to use this?" they're always right. Because the kids who say that are the kids who really aren't going to use trig ever again in their lives, except maybe to scrape out a single math credit in college by taking a "College Algebra" course that's easier than the trig course they're taking in high school. The kids who say "so fucking what?" are the kids who are going to be business people, car mechanics, policemen, English teachers, coaches, corporate trainers, office workers, journalists, politicians, plumbers, soldiers, and grocery store managers. You know, the 98% of the planet that doesn't use trigonometry. There really isn't an answer you can give them. You can convince them that trig is essential for making video games, curing cancer, and any number of things they care deeply about, but what they're really thinking is, "I'm not going to be making video games. I'm going to be designing the ad campaign for the video game," or, "I'm not going to be curing cancer; I'm going to be mopping the floors at the hospital." Or, "I'm going to work for my dad for a few years at the dealership and then run for city council, and I'm going to pass a $5 million bond to hire an urban planning agency that probably employs some people who know trig."
I don't know how you motivate those kids to learn trigonometry, but you can't do it by convincing them they will apply trig in the real world, unless you are a very gifted liar ;-)
I have to ask, do you realize how much of an outlier you are and how irrelevant your experience is to teaching normal kids mathematics? I mean... you immediately saw value in mathematically defining the relationship between the angle and the length. Teachers don't worry about kids like you (and me, and most of HN probably) because we'll be fine no matter how the class is taught. Wondering about how to get kids like you or me interested in math is like wondering how to get a cat interested in mice, or a Jersey Shore cast member interested in taking his shirt off. Teachers don't waste a second's thought on kids who will flourish regardless of the classroom environment. They worry about the marginal kids who might succeed if taught well but will flounder if taught poorly. A kid who really says "so fucking what?" to trigonometry is going to say "so fucking what?" to your trebuchet example, too. In the unlikely event he gives a damn about the length of the elastic rubber, he can just crank the attachment up and down and watch how the length changes, so who needs math?
The "best students" have sat through enough math classes that they know the right default assumption is that whatever is taught is probably useful. They get that engineering and science (including social science) depend on math, and math builds on previous math, and the high school curriculum doesn't have room for stuff that never comes up again later (okay, with the exception of geometry class, but even that is mostly useful.) Heck, even the "pretty good" students understand that. Those are the kids who get excited about real-world applications of trig: the kids who already get it and never ask "so fucking what?" Plus there's another group of kids who will never seriously ask "so fucking what," the ones who care about grades and college applications. They already know how they're going to use math: they're going to use it to get good grades in math class.
When students in trigonometry class really do say "so fucking what?" and "When are we ever going to use this?" they're always right. Because the kids who say that are the kids who really aren't going to use trig ever again in their lives, except maybe to scrape out a single math credit in college by taking a "College Algebra" course that's easier than the trig course they're taking in high school. The kids who say "so fucking what?" are the kids who are going to be business people, car mechanics, policemen, English teachers, coaches, corporate trainers, office workers, journalists, politicians, plumbers, soldiers, and grocery store managers. You know, the 98% of the planet that doesn't use trigonometry. There really isn't an answer you can give them. You can convince them that trig is essential for making video games, curing cancer, and any number of things they care deeply about, but what they're really thinking is, "I'm not going to be making video games. I'm going to be designing the ad campaign for the video game," or, "I'm not going to be curing cancer; I'm going to be mopping the floors at the hospital." Or, "I'm going to work for my dad for a few years at the dealership and then run for city council, and I'm going to pass a $5 million bond to hire an urban planning agency that probably employs some people who know trig."
I don't know how you motivate those kids to learn trigonometry, but you can't do it by convincing them they will apply trig in the real world, unless you are a very gifted liar ;-)
Apostle's textbook sounds interesting, not that I read math textbooks for fun (or any reason at all if I can avoid it).
We were taught (retaught) Calculus in first ("freshman") year, first semester Pure Math using Spivak's textbook which starts from a set of axioms that define the Real numbers. While this is anything but historical, I think this is better because you aren't "explaining away" stuff. At no point were we asked to simply accept something (once the axioms have been introduced and exercised and, I guess, setting aside the rigors of language and logic).
We were initially taught Calculus (in high school) the way you describe -- series and then limits -- and it royally sucked (no-one understood it and it provided no historical context).
We were taught (retaught) Calculus in first ("freshman") year, first semester Pure Math using Spivak's textbook which starts from a set of axioms that define the Real numbers. While this is anything but historical, I think this is better because you aren't "explaining away" stuff. At no point were we asked to simply accept something (once the axioms have been introduced and exercised and, I guess, setting aside the rigors of language and logic).
We were initially taught Calculus (in high school) the way you describe -- series and then limits -- and it royally sucked (no-one understood it and it provided no historical context).
Apostol starts similarly, with the first chapter mostly being really basic set theory (and discussion of the reals, couple of proofs about fields, etc).
Reading both volumes of Apostol is the sort of math introduction that any college student at all interested in math should have.
Reading both volumes of Apostol is the sort of math introduction that any college student at all interested in math should have.
I think part of the problem is how we are tested. I fell into the same situation in high school where I'd be able to derive complex formulas on the fly that I required to solve a problem, but it took me so long to do that compared to knowing the equation that I'd never finish exams (time would run out) and I had to resort to rote memorization in the end.
One way to level this is to allow cheat sheets or open books. This is a bold way of saying: "If you truly understand the topic, you'll be able to solve this problem even though it requires a mental step ahead from what we've seen in class." What usually happens here is that students who don't have a good grasp and get by on pattern-matching exam questions to questions done in class, complain about not having seen that type of question before.
Guess what, in real life, you almost always hit a type of question you haven't seen before! The kinds that have been known already are easily findable in any textbook. So why are schools focusing so much on the rote memorization / pattern matching rather than focusing on raw problem-solving skills? Algebra in 9th and 10th grade, precalc in 11th grade and intro calculus in 12 doesn't do anyone any good.
One way to level this is to allow cheat sheets or open books. This is a bold way of saying: "If you truly understand the topic, you'll be able to solve this problem even though it requires a mental step ahead from what we've seen in class." What usually happens here is that students who don't have a good grasp and get by on pattern-matching exam questions to questions done in class, complain about not having seen that type of question before.
Guess what, in real life, you almost always hit a type of question you haven't seen before! The kinds that have been known already are easily findable in any textbook. So why are schools focusing so much on the rote memorization / pattern matching rather than focusing on raw problem-solving skills? Algebra in 9th and 10th grade, precalc in 11th grade and intro calculus in 12 doesn't do anyone any good.
It's worth pointing out that Boaler's work is fairly controversial. There's a review by Bishop, Milgram, and Clopton that refutes many of her claims. Her work is interesting, but it seems that much of its appeal is that it confirms people's suspicions that the way they learned math was wrong.
To be fair, I don't know what the right way is. A lot of very different approaches have produced successful mathematicians. How to teach math is a recurring controversy in mathematics, and a scan of the Bulletins of the AMS will convince one that mathematicians are working on the same gut level that everyone else is, unfortunately.
To be fair, I don't know what the right way is. A lot of very different approaches have produced successful mathematicians. How to teach math is a recurring controversy in mathematics, and a scan of the Bulletins of the AMS will convince one that mathematicians are working on the same gut level that everyone else is, unfortunately.
After Vector Calc, I wanted to go back to the fundamentals, to understand instead of remembering.
I came across Silvanus Thompson's 1910 reprinted textbook Calculus Made Easy [1], and it was hands down the best primer on any topic I've delved into.
1. http://www.amazon.co.uk/Calculus-Made-Easy-Very-Simplest-Int...
I came across Silvanus Thompson's 1910 reprinted textbook Calculus Made Easy [1], and it was hands down the best primer on any topic I've delved into.
1. http://www.amazon.co.uk/Calculus-Made-Easy-Very-Simplest-Int...
I've Silvanus sitting in my shelf, but am yet to look into it yet.
A couple of recommendations (not specific to just Calculus):
- What is Mathematics? (Courant http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-...)
- Calculus (Apostle http://www.amazon.com/Calculus-Vol-One-Variable-Introduction...).
- Mathematics from the Birth of Numbers (http://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg...) This book was written by a Swedish surgeon without any background in Mathematics. He started working on this when his son started attending university. A recommended read.
- The Calculus Lifesaver (Adrian Banner). This book is supposed to be a guide for students to crack their exams. But I found the book surprisingly informative. http://press.princeton.edu/titles/8351.html
- Godel Escher Bach. I've read only the first couple of chapters. My interest in mathematics was rekindled to a great degree by Godel and the Incompleteness Theorem. (http://en.wikipedia.org/wiki/Kurt_G%C3%B6del#The_Incompleten...)
- http://us.metamath.org/. The concept alone makes me happy! Metamath is a collection of machine verifiable proofs. It uses ZFG to use prove complicated proofs by breaking it down to the most basic axioms. The fundamental idea is substitution - take a complicated proof, substitute it with valid expressions from a lower level and keep at it. It introduced me to ZFG and after wondering why 'Sets' were being taught repeatedly over the course of years when the only useful thing I found was Venn diagrams and calculating intersection and union counts, I finally understood that Set theory underpins Mathematical logic and vaguely how.
- The Philosophy of Mathematics. From the wiki: studies the philosophical assumptions, foundations, and implications of mathematics. It helped me understand how Mathematics is a science of abstractions. It finally validated the science as something that could be interesting and creative. http://plato.stanford.edu/entries/philosophy-mathematics/
I think the Philosophy of Mathematics should be taught during undergraduate courses that has Maths. It helps the students understand the nature of mathematics (at least the debates about it), which is usually pretty fuzzy for everyone.
A couple of recommendations (not specific to just Calculus):
- What is Mathematics? (Courant http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-...)
- Calculus (Apostle http://www.amazon.com/Calculus-Vol-One-Variable-Introduction...).
- Mathematics from the Birth of Numbers (http://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg...) This book was written by a Swedish surgeon without any background in Mathematics. He started working on this when his son started attending university. A recommended read.
- The Calculus Lifesaver (Adrian Banner). This book is supposed to be a guide for students to crack their exams. But I found the book surprisingly informative. http://press.princeton.edu/titles/8351.html
- Godel Escher Bach. I've read only the first couple of chapters. My interest in mathematics was rekindled to a great degree by Godel and the Incompleteness Theorem. (http://en.wikipedia.org/wiki/Kurt_G%C3%B6del#The_Incompleten...)
- http://us.metamath.org/. The concept alone makes me happy! Metamath is a collection of machine verifiable proofs. It uses ZFG to use prove complicated proofs by breaking it down to the most basic axioms. The fundamental idea is substitution - take a complicated proof, substitute it with valid expressions from a lower level and keep at it. It introduced me to ZFG and after wondering why 'Sets' were being taught repeatedly over the course of years when the only useful thing I found was Venn diagrams and calculating intersection and union counts, I finally understood that Set theory underpins Mathematical logic and vaguely how.
- The Philosophy of Mathematics. From the wiki: studies the philosophical assumptions, foundations, and implications of mathematics. It helped me understand how Mathematics is a science of abstractions. It finally validated the science as something that could be interesting and creative. http://plato.stanford.edu/entries/philosophy-mathematics/
I think the Philosophy of Mathematics should be taught during undergraduate courses that has Maths. It helps the students understand the nature of mathematics (at least the debates about it), which is usually pretty fuzzy for everyone.
If you've only read the first few chapters of Godel Escher Bach, you should really set a goal to continue reading. The book is filled with so much good information presented in a digestible format. Topics are slowly revealed throughout the book until you just get it. It's a great experience.
Available on Project Gutenberg here: http://www.gutenberg.org/files/33283/33283-pdf.pdf
I don't have a link handy, but there is a very clear scanned pdf of the second edition of Calculus Made Easy from 1914 available on archive.org
Have to agree on Calculus Made Easy. Found it in high school and quickly learned intro calculus better than anyone in the AP Calculus class.
> I could appreciate it a lot more when I understood what kind of problems were Newton and Leibniz trying to solve when they came up with the formalized notion of Calculus.
This always bothered me in many of my math classes. We'd go over a new formula or concept and just hammer it home until it stuck. Professors rarely gave real-world examples of the topics they were teaching or indicated to the students how these new theorem interacted with others. If I asked them explicitly I might get a more practical explanation, but it was rarely offered to me without inquiring first.
I used to have a lower opinion of people who'd complain about learning math because "it's so pointless, I'll never use any of this." It bothered me that they just didn't want to learn something new. In hindsight, even if that were partly true, I can't blame them for having that attitude because it may have been partially instilled in them by their professors.
I'll never forget the feelings I had after I connected the dots in my head and noticed the relationship between integrals. I knew that integration would give me the area underneath a curve, but now we were learning about double and triple integrals for surface area and volume. It dawned on me that I'm basically doing the same thing I was before, albeit in several planes with more curves. I remember a lot of different emotions: appreciation for the sensibility and beauty of mathematics; pride that I figured something out on my own; but also a tinge of frustration that this revelation was never encouraged by my professor.
This always bothered me in many of my math classes. We'd go over a new formula or concept and just hammer it home until it stuck. Professors rarely gave real-world examples of the topics they were teaching or indicated to the students how these new theorem interacted with others. If I asked them explicitly I might get a more practical explanation, but it was rarely offered to me without inquiring first.
I used to have a lower opinion of people who'd complain about learning math because "it's so pointless, I'll never use any of this." It bothered me that they just didn't want to learn something new. In hindsight, even if that were partly true, I can't blame them for having that attitude because it may have been partially instilled in them by their professors.
I'll never forget the feelings I had after I connected the dots in my head and noticed the relationship between integrals. I knew that integration would give me the area underneath a curve, but now we were learning about double and triple integrals for surface area and volume. It dawned on me that I'm basically doing the same thing I was before, albeit in several planes with more curves. I remember a lot of different emotions: appreciation for the sensibility and beauty of mathematics; pride that I figured something out on my own; but also a tinge of frustration that this revelation was never encouraged by my professor.
Exactly! You were fortunate enough to realize this during your course itself. But a vast majority of students never stumble upon this revealation and treat Maths as another chore that needs to be done with.
The insight you described is what Math classes should be about, not practicing problem after problem without any knowing what or why.
The insight you described is what Math classes should be about, not practicing problem after problem without any knowing what or why.
For those who have been inspired by this post and want to check out Apostol's excellent book, note that it is spelled that way. (It is a shortening of a Greek surname--Apostolopoulus, I think.)
Bibliography: https://en.wikipedia.org/wiki/Tom_M._Apostol.
Thanks for the correction. This mistake and the correction would be a good memory tool for me in the future!
Calculus was described by them using the concept of 'infinitesimals', not through Limits. Limits was a clever abstraction that was evolved later to better explain Calculus and keep it consistent.
It also isn't even necessary to use limits. Deriving calculus using the hyperreal number system (which has both infinites and infinitesimals) has been proved to be equivalent to calculus with limits (i.e., any theorem true in one is true in the other).
So if we want to, we could go back to using infinitesimals.
http://en.wikipedia.org/wiki/Hyperreal_number
It also isn't even necessary to use limits. Deriving calculus using the hyperreal number system (which has both infinites and infinitesimals) has been proved to be equivalent to calculus with limits (i.e., any theorem true in one is true in the other).
So if we want to, we could go back to using infinitesimals.
http://en.wikipedia.org/wiki/Hyperreal_number
Thanks for the link.
I wouldn't want to rigorously teach Calculus using *R unless the relevant proofs are reasonably accessible (which, given that we're talking Number Theory, seems like a remote possibility), but it's wonderful how much simpler they make the arguments if you work with them.
I wouldn't want to rigorously teach Calculus using *R unless the relevant proofs are reasonably accessible (which, given that we're talking Number Theory, seems like a remote possibility), but it's wonderful how much simpler they make the arguments if you work with them.
This is a good book. I found it preferable to regular epsilon delta when I was in highschool.
http://www.math.wisc.edu/~keisler/calc.html
http://www.math.wisc.edu/~keisler/calc.html
The relevant proofs are easily accessible, probably more so than the limit based proofs of regular calculus.
The only thing that isn't easily accessible is the proof of transference, namely that what's true for star-R (don't know how to make asterisks format nicely) is also true for R.
The only thing that isn't easily accessible is the proof of transference, namely that what's true for star-R (don't know how to make asterisks format nicely) is also true for R.
I mean the equivalence / transference proofs. Otherwise you're pulling a theorem out of the air and telling students to just accept it. And by "accessible" I mean easy enough to understand that they don't become a huge distraction (or worse).
If one starts with star-R to begin with, then why does one need the transference theorems?
You're then discussing math in terms of a set whose properties don't obviously match anything the student has dealt with. It needs to be established, one way or another, that adding infinitesimals etc. is not introducing new behavior.
I don't see why that can't just be stated. Students need somewhere to start from, and there's plenty already that we say "trust us for now, we'll prove it later" - and it's not like the reals actually match anything the student has dealt with, at the corner cases, either...
No, it means you are starting with a different (but equivalent) set of axioms.
Our use of R rather than star-R is merely a quirk of history - we managed to justify R to ourselves first, hence it's the "standard".
Our use of R rather than star-R is merely a quirk of history - we managed to justify R to ourselves first, hence it's the "standard".
You're working with strictly _more_ axioms, and the extra axioms don't seem justified unless you use the equivalence proof to show you haven't added undesired behavior.
(If you look at the other axioms, the hardest one to justify is the bounding axiom, and its necessity is reinforced by proof. Every other axiom fits in with the student's understanding of arithmetic.)
(If you look at the other axioms, the hardest one to justify is the bounding axiom, and its necessity is reinforced by proof. Every other axiom fits in with the student's understanding of arithmetic.)
I'm not spending a lot on this story or comments, only skimmed yours too (sorry) but I notice that you object to an emphasis on rote learning.
although it's nice to have history and background the truth is that the math that's most useful to us is pure rote - things like the multiplication table, adding and subtracting numbers.
when people go into a subject, like 3d game design, and 'wish they paid more attention in (that part of) math' they don't wish they had a finer appreciation for the background: they wish they would just know the formulas when they need them so they don't have to stop and think.
it's like logical arithmetic. if it's second nature to you you can refactor or's and and's and xor's and not's into and out of parentheses in code faster. who cares about the finer stuff behind it.
I am not saying this as someone who has a great deal of rote in me. But the part that I do have has served me well. I bet if I had been forced to memorize a 100*100 multiplication table and addition, and subtraction table (10,000 members each) and did my multiplication, division, addition, and subtraction, two digits at a time instead of how I do them (like everyone else) it would have served me quite well in life.
not saying that that's a good use of the precious little time kids have between 6 and 14, but just saying that rote is extremely useful for a lot of key things. set-theoretical logic, probably not so much.
although it's nice to have history and background the truth is that the math that's most useful to us is pure rote - things like the multiplication table, adding and subtracting numbers.
when people go into a subject, like 3d game design, and 'wish they paid more attention in (that part of) math' they don't wish they had a finer appreciation for the background: they wish they would just know the formulas when they need them so they don't have to stop and think.
it's like logical arithmetic. if it's second nature to you you can refactor or's and and's and xor's and not's into and out of parentheses in code faster. who cares about the finer stuff behind it.
I am not saying this as someone who has a great deal of rote in me. But the part that I do have has served me well. I bet if I had been forced to memorize a 100*100 multiplication table and addition, and subtraction table (10,000 members each) and did my multiplication, division, addition, and subtraction, two digits at a time instead of how I do them (like everyone else) it would have served me quite well in life.
not saying that that's a good use of the precious little time kids have between 6 and 14, but just saying that rote is extremely useful for a lot of key things. set-theoretical logic, probably not so much.
"As someone who was taught the 'traditional' way of mathematics, can someone give a few pointers of de-programming myself from the traditional way that I was taught."
I tutor high school students in math and breaking the memorization, process based approach to math is usually pretty difficult. The biggest thing is to absolutely avoid memorizing or even looking at formulas. Instead, try and look at things graphically. A lot of formulas seem confusing and unintuitive but if you look at it on a graph, it makes a lot more sense.
A basic example is something like the distance formulas. Students often say "Gah I can never remember the distance formula. I need to memorize it before the test" because
d = sqrt( (x1 - x2)^2 + (y1 - y2)^2 ) seems complex and confusing.
But if you draw a right triangle, it's apparent that the distance formula is just finding the hypotenuse.
In a nutshell, focus on the why's and not the how's.
I tutor high school students in math and breaking the memorization, process based approach to math is usually pretty difficult. The biggest thing is to absolutely avoid memorizing or even looking at formulas. Instead, try and look at things graphically. A lot of formulas seem confusing and unintuitive but if you look at it on a graph, it makes a lot more sense.
A basic example is something like the distance formulas. Students often say "Gah I can never remember the distance formula. I need to memorize it before the test" because
d = sqrt( (x1 - x2)^2 + (y1 - y2)^2 ) seems complex and confusing.
But if you draw a right triangle, it's apparent that the distance formula is just finding the hypotenuse.
In a nutshell, focus on the why's and not the how's.
As an undergrad math major, I've just recently run into this problem as I begin to take graduate level math courses. In the past (i.e. through high school math and basic calculus) I have always had a deeper understanding of the math going on such that I didn't really need to memorize many formulas or rules because I could always "reinvent" them (at least the simpler ones such as cos^2(x) + sin^2(x)=1) if I needed them.
I have found this level of understanding far more difficult to attain recently. For example, this past semester I took a linear algebra course. Whenever I am given a definition or property, I typically try to prove/justify it to myself or at least figure out why it is interesting. I found in my linear algebra course that this was very difficult to do, since there were so many definitions that just seemed to be "dropped in", unexplained by the professor. Thus, I found myself memorizing formulas and theorems instead of having the deep understanding of them I was used to.
I may have gone off on a little bit of a tangent, seeing as this article relates more to basic mathematics, but I think the underlying problem is the same. I'm not really sure what the solution is in my case, but I would guess that if my professor were to devote more time to explaining the usefulness of some of the more abstract concepts (such as eigenvalues) I would probably feel more comfortable with the subject.
I have found this level of understanding far more difficult to attain recently. For example, this past semester I took a linear algebra course. Whenever I am given a definition or property, I typically try to prove/justify it to myself or at least figure out why it is interesting. I found in my linear algebra course that this was very difficult to do, since there were so many definitions that just seemed to be "dropped in", unexplained by the professor. Thus, I found myself memorizing formulas and theorems instead of having the deep understanding of them I was used to.
I may have gone off on a little bit of a tangent, seeing as this article relates more to basic mathematics, but I think the underlying problem is the same. I'm not really sure what the solution is in my case, but I would guess that if my professor were to devote more time to explaining the usefulness of some of the more abstract concepts (such as eigenvalues) I would probably feel more comfortable with the subject.
Parents whose own math education was more traditional believe [...] that the presence of weaker students [in mixed ability groups] will drag down the better ones.
These parents are so wrong. Nothing makes you better at math than to explain what you have understood, well what you think you have understood, to other students. It's a challenging exercise that will benefit the brightest (except for pretentious asses I guess).
These parents are so wrong. Nothing makes you better at math than to explain what you have understood, well what you think you have understood, to other students. It's a challenging exercise that will benefit the brightest (except for pretentious asses I guess).
Is this satire? I think this is completely backwards. You can intuit math, you can't intuit the king of England during the 1400s, or what the word "defenestrate" means (well I guess you can if you know a bit of French or Latin).
Everybody forms their own way of doing mathematics through the mind, but vocabulary and history are simply memorization.
Everybody forms their own way of doing mathematics through the mind, but vocabulary and history are simply memorization.
The title is a (depressing) quote from a student who apparently didn't understand enough math to understand the problem with their position.
I did very poorly in math throughout high school (and mediocre in college). I still am not that great, but I feel I have a far deeper understanding than your average college or high school "A" student. It has nothing to do with how smart I am. It has everything to do with the fact that I have been using math in "real world" (ok, virtual world) problems for the last decade.
You absolutely cannot "teach" math (beyond the elementary level) and expect students to fully understand how it works. Memorizing equations is not understanding.
You must learn math, the hard way. Give a student a relevant problem, and tell them to solve it. Don't tell them how to solve it, let them derive the solution on their own. It might take weeks in some cases, but I guarantee they will have a much better understanding and increase their general problem-solving ability in all of mathematics.
You absolutely cannot "teach" math (beyond the elementary level) and expect students to fully understand how it works. Memorizing equations is not understanding.
You must learn math, the hard way. Give a student a relevant problem, and tell them to solve it. Don't tell them how to solve it, let them derive the solution on their own. It might take weeks in some cases, but I guarantee they will have a much better understanding and increase their general problem-solving ability in all of mathematics.
> The other, less common scene appears much more chaotic. Groups of students sit around circular tables discussing how to solve a particular problem, or standing at the whiteboard arguing about the best way to proceed. The teacher moves around the room talking with the different groups in turn, making suggestions as to how to proceed, or pointing out possible errors in a particular line of reasoning the students are following. Occasionally, the teacher will call the entire group to order and ask one group to explain their solution to the rest of the class, or to give a short, mini-lecture about a particular concept or method
I had some classes that did this. In was invariably even less useful than "please memorize this formula", simply because of the social pressures.
I had some classes that did this. In was invariably even less useful than "please memorize this formula", simply because of the social pressures.
Thanks for the many interesting comments. I'll reply jointly here to a few of the comments, and invite further discussion on some points made in the submitted article that I'm curious about.
From a top-level comment:
http://news.ycombinator.com/item?id=4084559
As someone who was taught the 'traditional' way of mathematics, can someone give a few pointers of de-programming myself from the traditional way that I was taught? (Although maybe it won't be so hard since I feel like I've forgotten quite a bit)
From a second-level comment, which has already received some helpful replies:
http://news.ycombinator.com/item?id=4084426
After Vector Calc, I wanted to go back to the fundamentals, to understand instead of remembering.
There is a FAQ page on the Epsilon Camp site
http://www.epsiloncamp.org/FAQ.php
that includes some Frequently Asked Questions articles about learning mathematics for deeper understanding. The FAQ article "Problems versus Exercises"
http://www.epsiloncamp.org/faq/faq_1.php
relates to what kind of work to set for yourself to build deeper understanding, and the FAQ article "Learning Mathematics"
http://www.epsiloncamp.org/faq/faq_3.php
points to writings by various mathematicians, including the book Numbers and Geometry by John Stillwell, about how to appreciate mathematics as a deep, connected subject.
The submitted article mentioned "Numerous studies over the past thirty years have shown that when people of any age and any ability level are faced with mathematical challenges that arise naturally in a real-world context that has meaning for them, and where the outcome directly matters to them, they rapidly achieve a high level of competence. How high? Typically 98 percent, that's how high. I describe some of those studies in my book The Math Gene (Basic Books, 2000)." The most striking example of this that I remember from a news report was a Wall Street Journal series in the 1990s that followed two young men in an inner city ghetto, one who was a good high school student and the other who was a street criminal. The street criminal usually skipped high school, but happened to show up the day students could take one of the major standardized tests (probably the PSAT, if I remember correctly). The street criminal, who sold illegal drugs among other activities, scored just as well on the test as the more regularly attending student who had learned most of his mathematics from school lessons. That's a rather stark illustration of what's missing in school lessons for children who don't have an outside-of-school environment for learning mathematics.
http://www.ams.org/notices/200502/fea-kenschaft.pdf
The article also says that many students say, "You have to be willing to accept that sometimes things don't look like - they don't see that you should do them. Like they have a point. But you have to accept them." I wonder how that relates to the quotation attributed to John von Neumann,
http://en.wikiquote.org/wiki/John_von_Neumann
"Young man, in mathematics you don't understand things. You just get used to them."
And from a third-level comment:
http://news.ycombinator.com/item?id=4084865
I experience math (and programming) quite differently than learning a language or painting: Once I grasp a concept, I can use it. Before that, it's mostly useless to me.
I ask, because when I studied mathematics in school, I had a drive to understand the general principles first before I launched into working on my homework, while some of my classmates were successful--at least in the context of school--by working on the homework and DEVELOPING some level of understanding as they tried to figure out answers for the homework. (I was in a "tracked" mathematics class, taking algebra in eighth grade in an era when most Americans took algebra in tenth grade, if at all, and most of my classmates had parents who were engineers or medical doctors and could ask their parents for help at home if the school lessons were confusing, as they often were.) I also have a very strongly visual approach to grappling with mathematical problems. So when I first learned algebra, which was presented to me as a bunch of "Do this to the equation, and then do this" with little rationale, I found that very dissatisfying. Later in the school year, we learned about coordinate graphing of systems of equations in the Cartesian plane, and I remember thinking, "Why didn't you tell me this in the first place?" For historical reasons, and perhaps for reasons of what most learners consider most easy, usually purely procedural algebra for solving systems of two equations in two unknowns has been taught in school before graphing systems of equations in the coordinate plane. But for some learners, it would be easier and more accessible to reverse that order. What do you think about the issue of students working first according to instructions, to DEVELOP understanding a la the von Neumann quotation, versus getting the "big picture," perhaps explicitly visually, before working on problems.
I'll comment also that the approach taken to learning mathematics in school in most of the newly industrialized countries of east Asia and southeast Asia is plainly superior to the United States approach for at least two reasons:
1) the school textbooks in those countries explicitly encourage students to THINK about why a procedure will or will not work, and about how many different ways there might be to solve a problem, and
2) the school textbooks show multiple representations of most mathematical concepts, building from "concrete to pictorial to abstract" as in the Singapore Primary Mathematics series
http://www.singaporemath.com/Primary_Mathematics_US_Ed_s/39....
and the follow-up New Elementary Mathematics series
http://www.singaporemath.com/New_Elementary_Math_s/47.htm
which interleave arithmetic, number theory, geometry, and algebra in increasing depth and interconnection throughout all grade levels.
From a top-level comment:
http://news.ycombinator.com/item?id=4084559
As someone who was taught the 'traditional' way of mathematics, can someone give a few pointers of de-programming myself from the traditional way that I was taught? (Although maybe it won't be so hard since I feel like I've forgotten quite a bit)
From a second-level comment, which has already received some helpful replies:
http://news.ycombinator.com/item?id=4084426
After Vector Calc, I wanted to go back to the fundamentals, to understand instead of remembering.
There is a FAQ page on the Epsilon Camp site
http://www.epsiloncamp.org/FAQ.php
that includes some Frequently Asked Questions articles about learning mathematics for deeper understanding. The FAQ article "Problems versus Exercises"
http://www.epsiloncamp.org/faq/faq_1.php
relates to what kind of work to set for yourself to build deeper understanding, and the FAQ article "Learning Mathematics"
http://www.epsiloncamp.org/faq/faq_3.php
points to writings by various mathematicians, including the book Numbers and Geometry by John Stillwell, about how to appreciate mathematics as a deep, connected subject.
The submitted article mentioned "Numerous studies over the past thirty years have shown that when people of any age and any ability level are faced with mathematical challenges that arise naturally in a real-world context that has meaning for them, and where the outcome directly matters to them, they rapidly achieve a high level of competence. How high? Typically 98 percent, that's how high. I describe some of those studies in my book The Math Gene (Basic Books, 2000)." The most striking example of this that I remember from a news report was a Wall Street Journal series in the 1990s that followed two young men in an inner city ghetto, one who was a good high school student and the other who was a street criminal. The street criminal usually skipped high school, but happened to show up the day students could take one of the major standardized tests (probably the PSAT, if I remember correctly). The street criminal, who sold illegal drugs among other activities, scored just as well on the test as the more regularly attending student who had learned most of his mathematics from school lessons. That's a rather stark illustration of what's missing in school lessons for children who don't have an outside-of-school environment for learning mathematics.
http://www.ams.org/notices/200502/fea-kenschaft.pdf
The article also says that many students say, "You have to be willing to accept that sometimes things don't look like - they don't see that you should do them. Like they have a point. But you have to accept them." I wonder how that relates to the quotation attributed to John von Neumann,
http://en.wikiquote.org/wiki/John_von_Neumann
"Young man, in mathematics you don't understand things. You just get used to them."
And from a third-level comment:
http://news.ycombinator.com/item?id=4084865
I experience math (and programming) quite differently than learning a language or painting: Once I grasp a concept, I can use it. Before that, it's mostly useless to me.
I ask, because when I studied mathematics in school, I had a drive to understand the general principles first before I launched into working on my homework, while some of my classmates were successful--at least in the context of school--by working on the homework and DEVELOPING some level of understanding as they tried to figure out answers for the homework. (I was in a "tracked" mathematics class, taking algebra in eighth grade in an era when most Americans took algebra in tenth grade, if at all, and most of my classmates had parents who were engineers or medical doctors and could ask their parents for help at home if the school lessons were confusing, as they often were.) I also have a very strongly visual approach to grappling with mathematical problems. So when I first learned algebra, which was presented to me as a bunch of "Do this to the equation, and then do this" with little rationale, I found that very dissatisfying. Later in the school year, we learned about coordinate graphing of systems of equations in the Cartesian plane, and I remember thinking, "Why didn't you tell me this in the first place?" For historical reasons, and perhaps for reasons of what most learners consider most easy, usually purely procedural algebra for solving systems of two equations in two unknowns has been taught in school before graphing systems of equations in the coordinate plane. But for some learners, it would be easier and more accessible to reverse that order. What do you think about the issue of students working first according to instructions, to DEVELOP understanding a la the von Neumann quotation, versus getting the "big picture," perhaps explicitly visually, before working on problems.
I'll comment also that the approach taken to learning mathematics in school in most of the newly industrialized countries of east Asia and southeast Asia is plainly superior to the United States approach for at least two reasons:
1) the school textbooks in those countries explicitly encourage students to THINK about why a procedure will or will not work, and about how many different ways there might be to solve a problem, and
2) the school textbooks show multiple representations of most mathematical concepts, building from "concrete to pictorial to abstract" as in the Singapore Primary Mathematics series
http://www.singaporemath.com/Primary_Mathematics_US_Ed_s/39....
and the follow-up New Elementary Mathematics series
http://www.singaporemath.com/New_Elementary_Math_s/47.htm
which interleave arithmetic, number theory, geometry, and algebra in increasing depth and interconnection throughout all grade levels.
A few amusing anecdotes.
I had a lot of trouble learning to solve 2 equations in 2 variables because I did not see the point. Given any word problem that you were supposed to solve that way, I could solve it in my head. It wasn't until I was shown that I couldn't solve 3 equations in 3 variables that I realized that I needed to learn the boring way.
Secondly the single most useful thing that I did in school was try to generate a table of how likely it was to get various dice rolls when you rolled 4 6-sided dice and took the top 3. I learned a lot from that, and that sparked my interest in math.
Thirdly my biggest complaint about the way we teach stuff is that we present matrices and matrix multiplication with no context. It makes no sense to people. But if you know what a linear function, and realize that a matrix is just a way to write one down, then matrix multiplication turns out to be just function composition.
Just think how surprising the associative law is for matrix multiplication. I remember sitting there thinking, "How on Earth did anyone think it up, and see the associative law?" It becomes something you memorize because it makes no sense.
But the associative law always holds for function composition. Given three functions f, g, h and a thing they act on v, then by definition:
((f o g) o h)(v) = (f o g)(h(v)) = f(g(h(v))) = f((g o h)(v)) = (f o (g o h))(v)
Since matrix multiplication is just a way to write out function composition for a certain class of functions, it likewise must follow the associative law. THAT is how they thought it up!
I can hear the complaints already. "Oh, but this is too abstract for college students, they can't possibly understand this approach!" Bull. They can, and they do if you have the guts to present it this way. I've done it, with success.
I had a lot of trouble learning to solve 2 equations in 2 variables because I did not see the point. Given any word problem that you were supposed to solve that way, I could solve it in my head. It wasn't until I was shown that I couldn't solve 3 equations in 3 variables that I realized that I needed to learn the boring way.
Secondly the single most useful thing that I did in school was try to generate a table of how likely it was to get various dice rolls when you rolled 4 6-sided dice and took the top 3. I learned a lot from that, and that sparked my interest in math.
Thirdly my biggest complaint about the way we teach stuff is that we present matrices and matrix multiplication with no context. It makes no sense to people. But if you know what a linear function, and realize that a matrix is just a way to write one down, then matrix multiplication turns out to be just function composition.
Just think how surprising the associative law is for matrix multiplication. I remember sitting there thinking, "How on Earth did anyone think it up, and see the associative law?" It becomes something you memorize because it makes no sense.
But the associative law always holds for function composition. Given three functions f, g, h and a thing they act on v, then by definition:
((f o g) o h)(v) = (f o g)(h(v)) = f(g(h(v))) = f((g o h)(v)) = (f o (g o h))(v)
Since matrix multiplication is just a way to write out function composition for a certain class of functions, it likewise must follow the associative law. THAT is how they thought it up!
I can hear the complaints already. "Oh, but this is too abstract for college students, they can't possibly understand this approach!" Bull. They can, and they do if you have the guts to present it this way. I've done it, with success.
"It is my experience that proofs involving matrices can be shortened by 50% if one throws the matrices out."
(Emil Artin, Geometric Algebra)
Don't you have to basically prove associativity to show that matrix multiplication is a function composition?
Not at all.
Start with linear transformation F from vector space V to vector space W, each of which has a basis (v1, ..., vn) and (w1, ..., wn). The matrix M representing F in that pair of bases is (F(v1) F(v2) ... F(vn)) where each vector is written as a column.
Given those bases it is easy to demonstrate that every linear function can be uniquely represented that way.
Thanks to the properties of linearity, it is easy to demonstrate that the special case of matrix multiplication of a matrix against a column is the same as applying that linear function to the corresponding vector. Furthermore you can demonstrate that given the basis and the matrix, you have actually defined a linear function. (Therefore completing the demonstration that matrices are a notation for linear functions, and linear functions are what matrices represent.)
With that in mind the matrix representing (F o G) is going to be ((F o G)(v1) ... (F o G)(vn)). And when you unwind that definition you find that function composition turns into matrix multiplication. (As long as all of the bases match up of course, don't forget them!)
At this point you now have a rule for matrix multiplication. Thanks to the correspondence to linear functions, you can derive all of its algebraic properties (including associativity) from the corresponding properties of linear functions.
Incidentally by keeping track of the role of the basis throughout the presentation, you make it much easier to work out change of basis matrices later. Which has a lot of potential to be confusing because they work out to be the inverse of what you'd naively guess them to be. For instance if you rotate your basis 30 degrees clockwise, the change of basis matrix you get is a rotation 30 degrees counter-clockwise. (This happens for the same reason that while you spin clockwise, it looks to you like the world is spinning counter-clockwise.)
So how do you get the change of basis matrix? Well, go back to the definition. Make your function be the identity (everything remains the same, and then you just write out a matrix which has each column being, in the new basis, the coordinates of the basis vectors for the old basis.
Now an exercise to demonstrate to yourself that you really understood this. Let V be the vector space of polynomials of degree at most 2, and W be the vector space of polynomials of degree at most 1. Let F be the linear function called "differentiation". Start with a coordinate system on V which is just (p(0), p(1), p(2)) and a corresponding coordinate system on W which is just (p(0), p(1)). In that pair of coordinate systems, what matrix represents F?
If you can figure that out, you probably understood the whole thing. If not, well...
(Big hint. There is a different pair of coordinate systems in which you can easily write down the answer. Use that fact...)
Start with linear transformation F from vector space V to vector space W, each of which has a basis (v1, ..., vn) and (w1, ..., wn). The matrix M representing F in that pair of bases is (F(v1) F(v2) ... F(vn)) where each vector is written as a column.
Given those bases it is easy to demonstrate that every linear function can be uniquely represented that way.
Thanks to the properties of linearity, it is easy to demonstrate that the special case of matrix multiplication of a matrix against a column is the same as applying that linear function to the corresponding vector. Furthermore you can demonstrate that given the basis and the matrix, you have actually defined a linear function. (Therefore completing the demonstration that matrices are a notation for linear functions, and linear functions are what matrices represent.)
With that in mind the matrix representing (F o G) is going to be ((F o G)(v1) ... (F o G)(vn)). And when you unwind that definition you find that function composition turns into matrix multiplication. (As long as all of the bases match up of course, don't forget them!)
At this point you now have a rule for matrix multiplication. Thanks to the correspondence to linear functions, you can derive all of its algebraic properties (including associativity) from the corresponding properties of linear functions.
Incidentally by keeping track of the role of the basis throughout the presentation, you make it much easier to work out change of basis matrices later. Which has a lot of potential to be confusing because they work out to be the inverse of what you'd naively guess them to be. For instance if you rotate your basis 30 degrees clockwise, the change of basis matrix you get is a rotation 30 degrees counter-clockwise. (This happens for the same reason that while you spin clockwise, it looks to you like the world is spinning counter-clockwise.)
So how do you get the change of basis matrix? Well, go back to the definition. Make your function be the identity (everything remains the same, and then you just write out a matrix which has each column being, in the new basis, the coordinates of the basis vectors for the old basis.
Now an exercise to demonstrate to yourself that you really understood this. Let V be the vector space of polynomials of degree at most 2, and W be the vector space of polynomials of degree at most 1. Let F be the linear function called "differentiation". Start with a coordinate system on V which is just (p(0), p(1), p(2)) and a corresponding coordinate system on W which is just (p(0), p(1)). In that pair of coordinate systems, what matrix represents F?
If you can figure that out, you probably understood the whole thing. If not, well...
(Big hint. There is a different pair of coordinate systems in which you can easily write down the answer. Use that fact...)
Thanks for the detailed explanation. I was subconsciously thinking "Matrix multiplication is this weird operation, which happens to be isomorphic to function composition in the space of linear transformations." rather than "Function composition, when functions are represented as matrices, is called matrix multiplication."
It sounds so obvious now...
It sounds so obvious now...
I need to stay silent until the end of the year to reflect how I missed that. Adios
“… try to generate a table of how likely it was to get various dice rolls when you rolled 4 6-sided dice and took the top 3.”
I felt the urge to code this. Here is the result: https://gist.github.com/2899137. It doesn’t tell you “likelihood of various dice rolls”; it can either print out the rolls for each trial or tell you how common each of the 6 numbers were in all the rolls.
I felt the urge to code this. Here is the result: https://gist.github.com/2899137. It doesn’t tell you “likelihood of various dice rolls”; it can either print out the rolls for each trial or tell you how common each of the 6 numbers were in all the rolls.
That's simulating the answer; you can get an exact answer without using a computer.
That’s true. But I thought doing it by hand would require writing a tediously large table because you have 6^3 possible roll results to give the probability of, if you were actually going to write the “likelihood of various dice rolls”. I suppose the appropriate compromise is a symbolic manipulation program like Mathematica, which can work with exact numbers easily while automating the creation of the table. (If anyone can explain the problem, it would be great if they could link to a document demonstrating the solution on somewhere like http://www.mathics.net/ .) Or is there an easier, simpler way to solve this by hand?
There are 6^4 possible rolls. But you don't need to list them all. You can take shortcuts.
After doing it by hand, I became interested in how I could do it by computer... :D
After doing it by hand, I became interested in how I could do it by computer... :D
http://anydice.com is pretty much your one stop shop anytime this comes up. It's pretty much the greatest thing ever if you're a P&P RPGer.
in this case: output [highest 3 of 4d6]
in this case: output [highest 3 of 4d6]
Thank you. That helps me understand matrices better!
Great comment -- chiming in because intuitive math (or the lack thereof) is a hot-button issue for me :).
I really dislike the von Neumann quote "Young man, in mathematics you don't understand things. You just get used to them."
I know what he's saying (there are some concepts you just need to internalize), but taken at face value it implies you stop looking for insights once you've "gotten used to it" (vs really grokking it).
I think learning is a spiral of theory & practice, i.e., present some principles, explain with examples, present more principles, explain with deeper examples, and so on.
Shameless plug, but check out this article on imaginary numbers:
http://betterexplained.com/articles/a-visual-intuitive-guide... HN discussion: http://news.ycombinator.com/item?id=2712575
I try to explain imaginaries by starting with negative numbers (something we're familiar with, but was counter intuitive at the time) and building up with examples (3 cows - 4 cows is "absurd", right? sqrt(-1) is "absurd", right?).
I don't think you can just define "i" as sqrt(-1) and give a bunch of problems, or talk about abstract visualizations for pages with any meaty examples: it's an interleaving spiral of both.
I really dislike the von Neumann quote "Young man, in mathematics you don't understand things. You just get used to them."
I know what he's saying (there are some concepts you just need to internalize), but taken at face value it implies you stop looking for insights once you've "gotten used to it" (vs really grokking it).
I think learning is a spiral of theory & practice, i.e., present some principles, explain with examples, present more principles, explain with deeper examples, and so on.
Shameless plug, but check out this article on imaginary numbers:
http://betterexplained.com/articles/a-visual-intuitive-guide... HN discussion: http://news.ycombinator.com/item?id=2712575
I try to explain imaginaries by starting with negative numbers (something we're familiar with, but was counter intuitive at the time) and building up with examples (3 cows - 4 cows is "absurd", right? sqrt(-1) is "absurd", right?).
I don't think you can just define "i" as sqrt(-1) and give a bunch of problems, or talk about abstract visualizations for pages with any meaty examples: it's an interleaving spiral of both.
Square root is a fundamentally geometric idea: you need to have a concept of area before you can find the ratio of an area to the side of a square. Much better than talking about what is or isn’t absurd in terms of pure symbols (like √-1 or whatever) is to give some geometric motivation of the definition of area in terms of orthogonality, which brings in the link to rotation, and finally results in complex numbers. By embedding complex numbers in so-called “geometric algebra”, you put them in their proper coherent and comprehensible context. http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
Can you recommend an introduction to geometric algebra & calculus? (preferably online) I've quickly read some but they were vague and in general quite unlike other mathematical texts. For example the document you linked to doesn't actually define the operations it uses. You can sort of conjure up a definition by piecing various statements together (by using the laws plus the operations' behavior on basis elements), but surely there must be a clearer introduction available? Other introductions were just plain wrong, for example one claimed that the curl of the curl is always 0. All that I've seen were strongly trying to persuade the reader of the awesomeness of GA rather than teaching it.
Hi jules, send me an email (see profile), and I’ll hook you up with some PDFs of books/papers.
I don't think von Neumann was saying "some concepts you just need to internalize" at all; his idea was more along the lines of
"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" (Jerry Bona)
In other words, intuitive notions of "understanding" aren't necessarily useful when dealing with "pure" mathematical ideas that lack real-world antecedents. (N.B.: the axiom of choice, the well-ordering principle, and Zorn's lemma are equivalent)
"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" (Jerry Bona)
In other words, intuitive notions of "understanding" aren't necessarily useful when dealing with "pure" mathematical ideas that lack real-world antecedents. (N.B.: the axiom of choice, the well-ordering principle, and Zorn's lemma are equivalent)
Thank you, I really love this kind of thing (former physics major here who sadly, I suppose, went through college never really peering deep into the meaning of tools I used on a daily basis).
I read a few of your articles before. You rock!
An anecdote from my father who helped 'slow' kids learn.
One time he dealt with a kid who struggled with maths. But he happened to know that this same child went to the horse races with his Dad and could calculate the pay-off from fairly complicated trifecta bets in his head.
Over and over my father found it was only motivation that was missing, not brain-power.
Over and over my father found it was only motivation that was missing, not brain-power.
Fairly common story, buddies that were quitting school around 15 couldn't compute percentages, yet they did all kind of crazy calculation on the fly with their own non-regular currency unit system to sell drugs. I couldn't follow any of them.
More than motivation (indeed money is a strong one) they had a purpose and a concrete/tangible object to reason about. You can and will test your ideas otherwise you will experience failure. Nobody make mistakes twice when they sell.
More than motivation (indeed money is a strong one) they had a purpose and a concrete/tangible object to reason about. You can and will test your ideas otherwise you will experience failure. Nobody make mistakes twice when they sell.
Actually money was not the main motivation here, as far as I am aware. It was more the bonding with and emulation of his Dad that motivated him.
No doubt the money helped if they won.
No doubt the money helped if they won.
Could change your handle?
Seriously, it's one of the most insulting Internet-names I know-of and I automatically vote down everything I see by you.
If you do good stuff, I suggest you don't begin every single post sneering at us. If you really need reminding, your name is a clever way of saying "I'm smart, you're dumb".
Maybe I'm really just a crank but in a perhaps illusory fashion I think I'm a moderately serious hn commentator.
And oddly enough, an adult.
And sure, I'm happy to be voted down to oblivion for this but hey, just keep this in mind.
Please, tweak your presentation. It would make totally anonymous and irrelevant me happy. I'm sure that doesn't mean anything.
Seriously, it's one of the most insulting Internet-names I know-of and I automatically vote down everything I see by you.
If you do good stuff, I suggest you don't begin every single post sneering at us. If you really need reminding, your name is a clever way of saying "I'm smart, you're dumb".
Maybe I'm really just a crank but in a perhaps illusory fashion I think I'm a moderately serious hn commentator.
And oddly enough, an adult.
And sure, I'm happy to be voted down to oblivion for this but hey, just keep this in mind.
Please, tweak your presentation. It would make totally anonymous and irrelevant me happy. I'm sure that doesn't mean anything.
Could change your handle?
Would you change yours if I changed mine? My screen name here comes from another site, where it was mandatory to use screen names (I'm actually more accustomed to online interaction with my real name and other people who use their real names), and then I carried it over to another site I heard about on the first site, and then over to here. I chose the screen name on the original site (the Art of Problem Solving site) because literally there a great many participants are young people below the age of majority, and it was useful there to distinguish who was an adult. At the second site where I used this screen name, I was again among users of mixed ages, with many minors participating, and it seemed helpful to keep this same screen name. On both of those online forums, most participants have correctly picked up the implication that "token" is part of my screen name to remove offense from the "adult" part of my screen name, by self-deprecation.
http://en.wikipedia.org/wiki/Self-deprecation
http://en.wikipedia.org/wiki/Tokenism
I kept the screen name here (although I could just as well have used my real name) simply by Occam's Razor, not desiring to choose a new screen name in case anyone recognized me from AoPS over here.
I automatically vote down everything I see by you.
I had no idea I have such a persistent fan club. I upvote based on the content of comments rather than on the screen name of the person commenting. I'm happy to hear from other users here any time they think I have given offense. I beg your pardon for any offense I have inadvertently caused by my choice of screen name here back 1300 days ago (just shortly before you came on board, it appears). I cherish the opportunity to learn from you and from other participants, and look forward to seeing your further comments.
P.S. Previous HN polls about participant ages suggest the age range here is almost entirely above the age of majority but almost entirely below my age. I have commented on the lack of sure-fire assurance that anyone here knows the full details of the age distribution here.
http://news.ycombinator.com/item?id=2322237
P.P.S. I'm sure there are other examples, but my family is the only example I know of personally of two generations both participating on HN. My adult son is rather busy with his work this summer, but he has posted here from time to time and often exceeds me in average comment karma score, I think.
Would you change yours if I changed mine? My screen name here comes from another site, where it was mandatory to use screen names (I'm actually more accustomed to online interaction with my real name and other people who use their real names), and then I carried it over to another site I heard about on the first site, and then over to here. I chose the screen name on the original site (the Art of Problem Solving site) because literally there a great many participants are young people below the age of majority, and it was useful there to distinguish who was an adult. At the second site where I used this screen name, I was again among users of mixed ages, with many minors participating, and it seemed helpful to keep this same screen name. On both of those online forums, most participants have correctly picked up the implication that "token" is part of my screen name to remove offense from the "adult" part of my screen name, by self-deprecation.
http://en.wikipedia.org/wiki/Self-deprecation
http://en.wikipedia.org/wiki/Tokenism
I kept the screen name here (although I could just as well have used my real name) simply by Occam's Razor, not desiring to choose a new screen name in case anyone recognized me from AoPS over here.
I automatically vote down everything I see by you.
I had no idea I have such a persistent fan club. I upvote based on the content of comments rather than on the screen name of the person commenting. I'm happy to hear from other users here any time they think I have given offense. I beg your pardon for any offense I have inadvertently caused by my choice of screen name here back 1300 days ago (just shortly before you came on board, it appears). I cherish the opportunity to learn from you and from other participants, and look forward to seeing your further comments.
P.S. Previous HN polls about participant ages suggest the age range here is almost entirely above the age of majority but almost entirely below my age. I have commented on the lack of sure-fire assurance that anyone here knows the full details of the age distribution here.
http://news.ycombinator.com/item?id=2322237
P.P.S. I'm sure there are other examples, but my family is the only example I know of personally of two generations both participating on HN. My adult son is rather busy with his work this summer, but he has posted here from time to time and often exceeds me in average comment karma score, I think.
I am sorry to speak in frustration. You know the thing about how humor, ever self-deprecating humor, doesn't translate over the Internet.
I'm sure you're joking but I would be willing to switch users name if you were willing.
I'm sure you're joking but I would be willing to switch users name if you were willing.
Thanks, Joe, I didn't see a way to reply to you off-forum in your user profile, so I'll follow up here. I sent pg the link to the subthread opened by your grandparent comment to your comment here, and asked him if the HN software allows changing a username with preservation of records of user submissions and comments. I told him I'd be happy to change my user name (and suggested a new one) if the HN software makes that possible. The software is not configured that way, pg told me, so I guess after 1301 days here I'm at risk to my reputation with this same user name I started with. On his part, pg also told me he didn't think my user name is a problem for the HN forum, but I appreciate your concern, and I thank you for drawing it to my attention. See you in the threads. (P.S. I looked up some of your previous comments in the same threads where I've commented before, by site-restricted Google search, and I see we share some interests and you have a thoughtful perspective on several issues I care about.)
Well thanks as well, I think I am over any annoyance here.
You are a gentleman. I think can only aspire to a similar level of politeness.
You are a gentleman. I think can only aspire to a similar level of politeness.
I hope you don't feel the need to change your user name, I can't see anything remotely offensive, to be honest. Your comments seem very well thought out and respectful also.
[deleted]
(Shrink the size of math text books. No reason why an algebra or trig text needs to be 900 pages, each page in primary colors)
I experienced the strict setting of sitting in a classroom, facing forwards, listening to the teacher give demonstrations and then answering tens of questions that followed the same pattern. This was from 16-18 and was only a small change from what I had experienced previously.
Before that we had at least been allowed to talk to each other, I think because the teacher recognised how needed that is.
Once I was studying for my A-levels however I was in a group of people who wanted to learn maths, we had chosen the subject after all. But we were given a teacher who demanded that we were silent, demanded that when we had a problem we asked her and not each other. Except for some students she came to the conclusion that their not understanding was their fault and would just tell them that they should have listened originally, for others it would just take too long to get an answer because the class was overcrowded with students. Twenty-five students at that level is far too many for one teacher.
Some students fell behind, and when they decided that instead of doing the parts of the subject they had no chance of understanding they would work on things they did understand to get up to speed, they had less advanced mock papers taken from them and told to do what they were told to do.
Students dropped maths, very able students dropped the subject. A lot of people's marks were hurt drastically due to just coming to hate the subject. We knew that discussing things with each other would help us a lot, it was good that I could turn to someone and ask them to explain something right then without having to wait for the single teacher to deal with the five other people who needed help before me. Hence, we did it anyway because it worked and we weren't stupid enough to keep silent just because the teacher told us to, we had to do this to learn, and we had voluntarily chosen the subject to learn.
So yes, I completely agree with the argument that the traditional approach is broken and that nobody wins. It's the only thing at that school that I have bad memories of, every other department, even other teachers within the maths department, knew the value of group based learning. A bad teacher using a bad method can ruin peoples view of an entire subject extremely quickly which can lead to disastrous results.
Once I was studying for my A-levels however I was in a group of people who wanted to learn maths, we had chosen the subject after all. But we were given a teacher who demanded that we were silent, demanded that when we had a problem we asked her and not each other. Except for some students she came to the conclusion that their not understanding was their fault and would just tell them that they should have listened originally, for others it would just take too long to get an answer because the class was overcrowded with students. Twenty-five students at that level is far too many for one teacher.
Some students fell behind, and when they decided that instead of doing the parts of the subject they had no chance of understanding they would work on things they did understand to get up to speed, they had less advanced mock papers taken from them and told to do what they were told to do.
Students dropped maths, very able students dropped the subject. A lot of people's marks were hurt drastically due to just coming to hate the subject. We knew that discussing things with each other would help us a lot, it was good that I could turn to someone and ask them to explain something right then without having to wait for the single teacher to deal with the five other people who needed help before me. Hence, we did it anyway because it worked and we weren't stupid enough to keep silent just because the teacher told us to, we had to do this to learn, and we had voluntarily chosen the subject to learn.
So yes, I completely agree with the argument that the traditional approach is broken and that nobody wins. It's the only thing at that school that I have bad memories of, every other department, even other teachers within the maths department, knew the value of group based learning. A bad teacher using a bad method can ruin peoples view of an entire subject extremely quickly which can lead to disastrous results.
> Numerous studies over the past thirty years have shown that when people of any age and any ability level are faced with mathematical challenges that arise naturally in a real-world context that has meaning for them, and where the outcome directly matters to them, they rapidly achieve a high level of competence. How high? Typically 98 percent, that's how high. I describe some of those studies in my book The Math Gene (Basic Books, 2000). I also provide an explanation of why those same people, when presented with the very same mathematical challenges in a traditional paper-and-pencil classroom fashion, perform at a lowly 37 percent level.
This. I love that this argument is being made. There is this notion among academics that the student should bend to the (fabricated?) stringent regulations on how math is taught or expressed. As a society, we accept the fact that there are those who grasp concepts, learn and develop sensibilities of material in various different ways.
For me personally, I found the way that math was taught in school to be completely disconnected with its purpose. There would be times where material was applied, but more often not. No other subject can get away with this. Music - instruments, scale and training. Art - painting, modelling and theory. English – writing, reading & comprehension. Science - hypothesis, experiments, conclusion. Most subjects have an execution factor. How far Jamie has to walk to get 3 bags of milk is not execution, it's practise. This is a gross simplification of a beautiful subject - but this is the point where many get lost: Purpose.
For those who absorb material differently, this is where the conversation needs to start.
This. I love that this argument is being made. There is this notion among academics that the student should bend to the (fabricated?) stringent regulations on how math is taught or expressed. As a society, we accept the fact that there are those who grasp concepts, learn and develop sensibilities of material in various different ways.
For me personally, I found the way that math was taught in school to be completely disconnected with its purpose. There would be times where material was applied, but more often not. No other subject can get away with this. Music - instruments, scale and training. Art - painting, modelling and theory. English – writing, reading & comprehension. Science - hypothesis, experiments, conclusion. Most subjects have an execution factor. How far Jamie has to walk to get 3 bags of milk is not execution, it's practise. This is a gross simplification of a beautiful subject - but this is the point where many get lost: Purpose.
For those who absorb material differently, this is where the conversation needs to start.
> How far Jamie has to walk to get 3 bags of milk is not execution
Fewer examples are that planted in reality, although you still encounter crossing trains and leaking bathtubs. But even if it's just about finding the reason of a geometric suite or calculating the derivative of some function, the teaching process is the same as in the Jamie example:
Math is not about driving screws with the screwdriver given to you, it's about understanding what a screwdriver is made of, why it is shaped this way, and building your own appropriate screwdriver should you encounter an unknown screw.
The current system is so hopelessly wrong that when I was faced with it my personal solution was to solve stuff in the quickest way possible so that I'm not bothered with such crap anymore. Incidentally to achieve this I taught myself to truly understand mathematics. The majority of others though could not be bothered and simply learned the rules by rote, because it was so complicated (and of course it is if you understand zilch about what you're doing). The irony is that the whole teaching process ended up this way because some people up above deemed mathematics too hard and "simplified" teaching. Then the whole thing got out of hand in a self-sustaining loop.
Fewer examples are that planted in reality, although you still encounter crossing trains and leaking bathtubs. But even if it's just about finding the reason of a geometric suite or calculating the derivative of some function, the teaching process is the same as in the Jamie example:
step 1: those are the Rules and Theorems.
step 2: IF $situation THEN apply $rule42
step 3: solved!
Math is not about driving screws with the screwdriver given to you, it's about understanding what a screwdriver is made of, why it is shaped this way, and building your own appropriate screwdriver should you encounter an unknown screw.
The current system is so hopelessly wrong that when I was faced with it my personal solution was to solve stuff in the quickest way possible so that I'm not bothered with such crap anymore. Incidentally to achieve this I taught myself to truly understand mathematics. The majority of others though could not be bothered and simply learned the rules by rote, because it was so complicated (and of course it is if you understand zilch about what you're doing). The irony is that the whole teaching process ended up this way because some people up above deemed mathematics too hard and "simplified" teaching. Then the whole thing got out of hand in a self-sustaining loop.
Painting, modelling and theory; writing, reading & comprehension. These are practice. Getting your painting into a gallery or your book into the hands of an unobligated reader - that is execution.
In Civil Engineering, we ran problem after problem in school. Nothing I do now professionally in civil design resembles the practice I did in school.
Which is just to say, execution should be the goal of ALL programs, but none seem to come close.
[edit] it's Friday morning and I'm done; any suggestions for a synonym for 'unobligated'?
In Civil Engineering, we ran problem after problem in school. Nothing I do now professionally in civil design resembles the practice I did in school.
Which is just to say, execution should be the goal of ALL programs, but none seem to come close.
[edit] it's Friday morning and I'm done; any suggestions for a synonym for 'unobligated'?
I agree that they are practice in some sense, but the end result is valuable immediately. When I was a young student, I would take my painting, science experiment or short story home to show my mom. When I did well in math I showed my grade, not the actual work. Here is where we find perception of activity vs. successful comprehension.
Math is not valuable unless it has purpose and if some are not given purpose they will stop investigating it. It's easy to categorize those who have that viewpoint. As someone who found math later in life on my own terms, I can say that it was much more enjoyable than what was being taught as a student.
This was foretold in each math class I had in the schools I attended. There was always a "Math is necessary for..." poster on the wall in the classroom, and yet, no other class had a "Art is necessary for..." or "History is necessary for..." poster. We've identified the problem. Now what do we do about it?
Language Studies (aside from immersion) also could benefit from being more execution-oriented. For me, 4 years of French class was easily lapped by a month in Quebec.
Math is not valuable unless it has purpose and if some are not given purpose they will stop investigating it. It's easy to categorize those who have that viewpoint. As someone who found math later in life on my own terms, I can say that it was much more enjoyable than what was being taught as a student.
This was foretold in each math class I had in the schools I attended. There was always a "Math is necessary for..." poster on the wall in the classroom, and yet, no other class had a "Art is necessary for..." or "History is necessary for..." poster. We've identified the problem. Now what do we do about it?
Language Studies (aside from immersion) also could benefit from being more execution-oriented. For me, 4 years of French class was easily lapped by a month in Quebec.
I completely agree. I found math later too, at 29, and had it been taught better it might have clicked with me earlier.
The only academic program I've experienced that got right to the crux of its execution was History: but only because I feel its only real purpose is self reflection.
The only academic program I've experienced that got right to the crux of its execution was History: but only because I feel its only real purpose is self reflection.
I found math interesting in school, but then I'm also virtually certain I was the only one in my high school classes actually reading the math textbooks. Even at an engineering college, I don't think reading the calculus textbook was the norm.
A shame, really. A lot of value ignored there.
A shame, really. A lot of value ignored there.
"[edit] it's Friday morning and I'm done; any suggestions for a synonym for 'unobligated'?"
"Voluntary" or "volunteer"?
"Voluntary" or "volunteer"?
Thanks, that almost works for me.
I like the notion of 'unobligated' in that it stresses that, for this instance, in school one's readers are obligated. With 'volunteer' I don't feel that condition is evoked; in fact, I can imagine people volunteering to read one's as yet unpublished work, but still feeling obligated to volunteer as a friend. But 'unobligated' is a mouth full of tongue to say, and if it should express freedom and willingness then it should lyrically sound that way. 'Volunteer' accomplishes that part, but leaves me wanting to tack on another qualifier.
Oh Friday morning.
I like the notion of 'unobligated' in that it stresses that, for this instance, in school one's readers are obligated. With 'volunteer' I don't feel that condition is evoked; in fact, I can imagine people volunteering to read one's as yet unpublished work, but still feeling obligated to volunteer as a friend. But 'unobligated' is a mouth full of tongue to say, and if it should express freedom and willingness then it should lyrically sound that way. 'Volunteer' accomplishes that part, but leaves me wanting to tack on another qualifier.
Oh Friday morning.
For me personally, I found the way that math was taught in school to be completely disconnected with its purpose. There would be times where material was applied, but more often not. No other subject can get away with this. Music - instruments, scale and training. Art - painting, modelling and theory. English – writing, reading & comprehension. Science - hypothesis, experiments, conclusion. Most subjects have an execution factor.
If science fair projects or English class writing assignments count as "execution" then certainly solving word problems counts as "execution." None of those things have real-world value, except in the very rare cases of extremely talented students who write a publishable story or produce valuable scientific results. The purpose of English class is to develop skills that will be applied outside English class, just like math.
What I personally saw in high school was that teachers were pedagogically obsessed with application in context, exactly what you call execution, but were at a loss as to how to apply the principle in practice. A project to create "context" for solving math problems could take many hours of class time and result in students using math only a handful of times. Solving a particular math problem is rather like hitting a curveball or playing a piece on the piano. You have relatively few chances to execute in a meaningful context -- a few dozen times per year, if you're lucky. Those natural contexts are just not sufficient for developing skill. So skills are developed in artificial contexts: hundreds of swings in batting practice, hours of practice alone at the piano, and many, many math problems with no real context.
It's easy to provide imaginary context, of course. I'm solving for the side of this triangle because I'm building a bridge and people could die (or I could get fired) unless I can figure out how long this side is. People don't find that very compelling, and that isn't unique to math. Kids taking batting practice or playing etudes aren't immersed in vivid major league or concert fantasies every time they swing or strike a key -- a lot of the time, they're fighting with their minds to fully engage with the task. But they take it for granted that they have to practice to succeed, while in math we have almost reached an attitude that practice is inimical to understanding. It's time we admitted that math is just the same as any other skill: little understanding can exist without competence, practice deepens understanding, and the mind is not freed to combine basic skills fluently until the basic skills become second nature.
It would be nice if it were possible to create a compelling context for every practice problem, but it isn't, not any more than you can create a compelling artistic context for every musical scale or writing exercise.
If science fair projects or English class writing assignments count as "execution" then certainly solving word problems counts as "execution." None of those things have real-world value, except in the very rare cases of extremely talented students who write a publishable story or produce valuable scientific results. The purpose of English class is to develop skills that will be applied outside English class, just like math.
What I personally saw in high school was that teachers were pedagogically obsessed with application in context, exactly what you call execution, but were at a loss as to how to apply the principle in practice. A project to create "context" for solving math problems could take many hours of class time and result in students using math only a handful of times. Solving a particular math problem is rather like hitting a curveball or playing a piece on the piano. You have relatively few chances to execute in a meaningful context -- a few dozen times per year, if you're lucky. Those natural contexts are just not sufficient for developing skill. So skills are developed in artificial contexts: hundreds of swings in batting practice, hours of practice alone at the piano, and many, many math problems with no real context.
It's easy to provide imaginary context, of course. I'm solving for the side of this triangle because I'm building a bridge and people could die (or I could get fired) unless I can figure out how long this side is. People don't find that very compelling, and that isn't unique to math. Kids taking batting practice or playing etudes aren't immersed in vivid major league or concert fantasies every time they swing or strike a key -- a lot of the time, they're fighting with their minds to fully engage with the task. But they take it for granted that they have to practice to succeed, while in math we have almost reached an attitude that practice is inimical to understanding. It's time we admitted that math is just the same as any other skill: little understanding can exist without competence, practice deepens understanding, and the mind is not freed to combine basic skills fluently until the basic skills become second nature.
It would be nice if it were possible to create a compelling context for every practice problem, but it isn't, not any more than you can create a compelling artistic context for every musical scale or writing exercise.
I experienced one really good example of application in high-school. We had one assignment that was coordinated between Science (in this case Biology) and English where we wrote a paper and the content was given a grade by the Biology teacher and the style/formatting, etc. was graded by the English teacher. It would have been nice if there were more things like this, as it directly addresses a concern pg expressed in one of his essays:
" Certainly schools should teach students how to write. But due to a series of historical accidents the teaching of writing has gotten mixed together with the study of literature. And so all over the country students are writing not about how a baseball team with a small budget might compete with the Yankees, or the role of color in fashion, or what constitutes a good dessert, but about symbolism in Dickens."
" Certainly schools should teach students how to write. But due to a series of historical accidents the teaching of writing has gotten mixed together with the study of literature. And so all over the country students are writing not about how a baseball team with a small budget might compete with the Yankees, or the role of color in fashion, or what constitutes a good dessert, but about symbolism in Dickens."
My school tried to adopt that approach. It was called "writing across the curriculum." It didn't last long at my school, which I thought was too bad, since it sounded like a pretty cool idea to me.
You may have point there. However, I experience math (and programming) quite differently than learning a language or painting: Once I grasp a concept, I can use it. Before that, it's mostly useless to me. Execercising it more afterwards does improve my usage, but more in the sense, that I'm quicker to spot situations where I can (or can't!) use that particular feature/theorem. Sometimes I happen to see a new angle which enables new tricks. Learning to play piano on the other hand is a lot about muscle memory, which you train by repeating the same thing again and again. Yes, there is part of this in math too, but thats the handicraft-stuff. Arithmetics. Things a computer can do better.
I recently picked up my neice's high school alegbra book and the "mile wide inch deep" problem is plainly evident: way too much clutter and convolving of topics. When I helped her with some problems sets, it became apparent that the teacher had been drilling the students with rigid mechanized approaches to solving different types of problems. Actual THINKING about the problem at hand and gaining intuitive grasp of graphing and relationships seems to be completely ignored in the teaching. Very sad indeed.
I was generally regarded as math prodigy. I got my first algebra book when I was 4 years old, and mastered the material quickly. I eventually got a mathematics PhD from Harvard when I was 19 years old.
Traditional mathematics classrooms contributed very little to that learning, except for the portion of a university education directed toward mathematics major. And I even have my doubts about that part.
Meanwhile, the best bit of math teaching I've ever done may have come when I was helping my stepdaughter with her algebra homework. I noticed her applying on over-specific rule to a class of formula simplification problems. So I deliberately changed the problem she was working on to one where the bogus rule didn't apply ... :)
I've come to believe that for students of all levels, the most important part of mathematics education is when you coach them through problem-solving. Yes, the formalities are wonderfully powerful and fun, at least to my taste. But they should be a last resort, whether for solving a problem or checking the soundness of a result. Formalities-first is exactly the wrong way to approach things, for math users and math teachers alike.
Traditional mathematics classrooms contributed very little to that learning, except for the portion of a university education directed toward mathematics major. And I even have my doubts about that part.
Meanwhile, the best bit of math teaching I've ever done may have come when I was helping my stepdaughter with her algebra homework. I noticed her applying on over-specific rule to a class of formula simplification problems. So I deliberately changed the problem she was working on to one where the bogus rule didn't apply ... :)
I've come to believe that for students of all levels, the most important part of mathematics education is when you coach them through problem-solving. Yes, the formalities are wonderfully powerful and fun, at least to my taste. But they should be a last resort, whether for solving a problem or checking the soundness of a result. Formalities-first is exactly the wrong way to approach things, for math users and math teachers alike.
I'm interested in how you mastered algebra at 4. Were you already strong at arithmetic when you got the algebra book (and algebra was just a continuation of your already acceleratd progress) or was algebra just very intuitive and the start of a great math career?
Can't speak for OP, but I started learning algebra at 5. It happened that I really liked playing with calculators at age 2-3, and that helped give me an extremely solid mental arithmetic foundation.
Essentially, the special thing about me probably wasn't some super-human neural circuitry for understanding math. It was the fact that I was wired, somehow, to enjoy playing with a calculator. I had more arithmetic practice by age 4 than most kids have by age 11.
Essentially, the special thing about me probably wasn't some super-human neural circuitry for understanding math. It was the fact that I was wired, somehow, to enjoy playing with a calculator. I had more arithmetic practice by age 4 than most kids have by age 11.
I feel like I can relate to this (a little). I'm no prodigy, but even as a young kid (6 or 7) I would write down algebra problems all day and solve them. I did a ton of mental math, too, writing down several dozen 2-digit numbers and adding them all in my head.
Haven't really thought about that in a long time. Playing with a calculator also got me into programming (TI-Basic on the TI-83 in middle school).
Haven't really thought about that in a long time. Playing with a calculator also got me into programming (TI-Basic on the TI-83 in middle school).
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>In an international survey conducted in 2003, students from forty countries were asked whether they agreed or disagreed with the statement: "When I study math, I try to learn the answers to the problems off by heart."
Is "off by heart" really a well known phrase in 40 countries? I'm from the US and this question really confused me (even though I'm aware of the phrase "by heart").
Is "off by heart" really a well known phrase in 40 countries? I'm from the US and this question really confused me (even though I'm aware of the phrase "by heart").
They obviously didn't ask this question in English in all these countries.
It is in the non-rebellious parts of the British Empire :-)
https://encrypted.google.com/search?q=I+try+to+learn+the+ans...
https://encrypted.google.com/search?q=I+try+to+learn+the+ans...
I'm all about physical metaphors being used to teach math - instead of handing someone an equation and saying that the equation is what the concept being taught is, it's easier to remember the physical metaphor that the concept is like, and then to see how the equation describes both the concept and the metaphor.
More of this: https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From
More of this: https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From
I'm a big fan of that approach. Betterexplained.com uses the method.
>In Math You Have to Remember, In Other Subjects You Can Think About It
This title/quotation is quite depressing, since the student has it totally backwards. Math and any field with mathematical roots are the only things you can think about and figure out. I can't count the number of times I walked into an EE test unprepared and was able to work out some formula, which I hadn't memorized, from other principles.
Most of my history/english/social studies consisted of rote memorization of facts, dates, and obscure words. I'm still flabbergasted on why the SAT has middle school level math problems while also testing you on an intractable memorization problem: vocabulary.
[update: apparently, analogies have been removed from the SAT. thanks, nickbarnwell]
This title/quotation is quite depressing, since the student has it totally backwards. Math and any field with mathematical roots are the only things you can think about and figure out. I can't count the number of times I walked into an EE test unprepared and was able to work out some formula, which I hadn't memorized, from other principles.
Most of my history/english/social studies consisted of rote memorization of facts, dates, and obscure words. I'm still flabbergasted on why the SAT has middle school level math problems while also testing you on an intractable memorization problem: vocabulary.
[update: apparently, analogies have been removed from the SAT. thanks, nickbarnwell]
> Most of my history/english/social studies consisted of rote memorization of facts, dates, and obscure words. I'm still flabbergasted on why the SAT has middle school level math problems while also testing you on an intractable memorization problem: vocabulary.
As someone who still has the SATs fresh in mind, vocabulary is actually a very small, and very easy, portion of the exam. It's a subcomponent of the Critical Reading section, which largely concerns itself with drawing conclusions from a given text. Anyone even moderately well-read should be capable of acing CR, and it was a not unusual occurence at my gymnasium (~5 out of 15 students in my year managed 800s, IIRC)
As someone who still has the SATs fresh in mind, vocabulary is actually a very small, and very easy, portion of the exam. It's a subcomponent of the Critical Reading section, which largely concerns itself with drawing conclusions from a given text. Anyone even moderately well-read should be capable of acing CR, and it was a not unusual occurence at my gymnasium (~5 out of 15 students in my year managed 800s, IIRC)
It's been about a decade since I took mine. Apparently, they got rid of analogies, which is what I was thinking of:
>In 2005, the test was changed again, largely in response to criticism by the University of California system.[32] Because of issues concerning ambiguous questions, especially analogies, certain types of questions were eliminated (the analogies from the verbal and quantitative comparisons from the Math section). http://en.wikipedia.org/wiki/SAT#2005_changes
That sounds like a great change. Each analogy required you to know 4 words (at least one of which would be rather obscure). I read quite a bit in grade school (still proud of myself for seemingly being 1 out of only 3 students who read the required 750 page John Adams over a summer), yet suffered miserably during this section.
>In 2005, the test was changed again, largely in response to criticism by the University of California system.[32] Because of issues concerning ambiguous questions, especially analogies, certain types of questions were eliminated (the analogies from the verbal and quantitative comparisons from the Math section). http://en.wikipedia.org/wiki/SAT#2005_changes
That sounds like a great change. Each analogy required you to know 4 words (at least one of which would be rather obscure). I read quite a bit in grade school (still proud of myself for seemingly being 1 out of only 3 students who read the required 750 page John Adams over a summer), yet suffered miserably during this section.
I'm (un)fortunate enough to have taken the SAT long enough ago to have had analogies on it. I suffered through them, scoring disproportionately poorly on them as compared to my math score. It was frustrating since I'm a voracious reader. I did not understand why such an important test would require memorizing so many obscure words to perform well, especially since most of the math section is about solving puzzles and trick questions. It seemed a completely useless skill.
It was years later that my (bilingual) wife explained her view on the analogies. She saw them as logical puzzles to figure out. If a student understands word roots and has some concept of foreign (romantic) languages or latin, it is simple for them to apply this knowledge to remove possible answers, figure out properties of the words, and solve the analogy.
My high school (even with 3 years of mostly worthless spanish) never presented me this toolbox. We spent 2 years in english classes with SAT prep vocabulary tests. We memorized random words and their definitions. It never occurred to me that this was contrary to the purpose of the verbal section of the test.
It is sad that people seem to feel the same way about the math section as I felt about the analogies section. "There's no way I can memorize every possible answer."
It was years later that my (bilingual) wife explained her view on the analogies. She saw them as logical puzzles to figure out. If a student understands word roots and has some concept of foreign (romantic) languages or latin, it is simple for them to apply this knowledge to remove possible answers, figure out properties of the words, and solve the analogy.
My high school (even with 3 years of mostly worthless spanish) never presented me this toolbox. We spent 2 years in english classes with SAT prep vocabulary tests. We memorized random words and their definitions. It never occurred to me that this was contrary to the purpose of the verbal section of the test.
It is sad that people seem to feel the same way about the math section as I felt about the analogies section. "There's no way I can memorize every possible answer."
I actually rather enjoyed the analogy section as they were like small logic puzzles and far more enjoyable than the incredibly dry passages they choose for the comprehension sections. Admittedly, I do enjoy esoteric vocabulary and have an unusually good memory for it.
With regard to the removal, I had thought it was due to complaints of cultural bias towards upper class whites; the canonical example something along like "oarsman : _______ as runner: marathon"
With regard to the removal, I had thought it was due to complaints of cultural bias towards upper class whites; the canonical example something along like "oarsman : _______ as runner: marathon"
As someone who was taught the 'traditional' way of mathematics, can someone give a few pointers of de-programming myself from the traditional way that I was taught? (Although maybe it won't be so hard since I feel like I've forgotten quite a bit)
Someone already recommended this for calculus which looks pretty promising: http://www.gutenberg.org/files/33283/33283-pdf.pdf
Someone already recommended this for calculus which looks pretty promising: http://www.gutenberg.org/files/33283/33283-pdf.pdf
If you can code, Project Euler is really good. Particularly the problems past #100; those tend to concentrate more on mathematical insights than programming ones. It's what kindled my interest in math in high school.
Yes, but Project Euler merely gives you a stimulus to go out and research, and a test to apply your new skills. They are not so much a tool on their to acquire the skills.
E.g. learning about dynamic programming will simplify at least half of all Project Euler problems. But it would be rather harder than necessary to try and come up with all the generalities of dynamic programming just from personal attempts at solving Project Euler problems.
E.g. learning about dynamic programming will simplify at least half of all Project Euler problems. But it would be rather harder than necessary to try and come up with all the generalities of dynamic programming just from personal attempts at solving Project Euler problems.
You probably don't need to deprogram anything.
A problem that you are motivated to solve is a good idea. Because it provides clear motivation. If such a thing isn't forthcoming, maybe think of understanding the concepts as the problem to solve (which is pretty wishy-washy, but it is likely going to be a difficult task if it isn't providing any satisfaction).
A problem that you are motivated to solve is a good idea. Because it provides clear motivation. If such a thing isn't forthcoming, maybe think of understanding the concepts as the problem to solve (which is pretty wishy-washy, but it is likely going to be a difficult task if it isn't providing any satisfaction).
Read `The Pleasure of Counting'. It's an excellent mathemical book for teenagers written by an actual mathematician and a gentle but real introduction to real mathematics. No dumbing down.
The substance of the article seems good, but why was it necessary to mention that it was a female high school student? Is her lack of a penis somehow involved with this discussion?
Uh... perhaps since we have huge gender gaps in STEM fields, such a quotation by a female high school student is especially alarming?
As mistercow says, it would have been a good idea to explain that. That detail just hanging there with no explanation caused me to pause as well, because I've encountered some mathematicians (not people I've worked with, just visiting professors and such) that felt the need to tell stories about person X, who was in some way not as clueful as they'd expected, "and by the way she was a woman."
I'm not in any way suggesting that the article author (Devlin) is like that, just that I would personally try not to make a statement like that without proactively making it clear that I'm not "that guy."
I'm not in any way suggesting that the article author (Devlin) is like that, just that I would personally try not to make a statement like that without proactively making it clear that I'm not "that guy."
Well, if the author had tied that point into the article in some way, that would be fine. But he didn't.
My favorite part about the benefits of learning to solve problems rather than memorizing skills for a particular section of a book at a time:
> When they had been at school, their social class, as determined by their parents' jobs, were the same at both schools. But eight years later, the young adults from Phoenix Park were working in more highly skilled or professional jobs than the Amber Hill adults... 65 percent of the Phoenix Park adults were in jobs more professional than their parents, compared with 23 percent of Amber Hill adults. In fact, 52 percent of Amber Hill adults were in less professional jobs than their parents, compared with only 15 percent of the Phoenix Park graduates...
Students at the school that that taught problem solving rather than memorization continued math education in college because they enjoyed it, leading 65% of them to get better jobs than their parents at only 24 years old. On the other hand, the majority of students at the traditional high school avoided math in college, and chose "less professional" jobs than their parents at 24. Over their lifespans, I can only imagine the disparity increased...
> Which statement would give you more pleasure? "Because of good teaching, my child scored 79% on her last math test," or "Because of good teaching, my child has a much better job and leads a far more interesting and rewarding life than me."
The author mentions that even the best students at traditional schools had trouble on tests since they practiced the skill-of-the-day on assignments, leading me to believe they would show remarkable improvement on standardized testing in a new curriculum. Maybe the author believes that testing students after-the-fact is not as good a use of taxpayer money as providing resources and education for teachers to change the curriculum would be.
> When they had been at school, their social class, as determined by their parents' jobs, were the same at both schools. But eight years later, the young adults from Phoenix Park were working in more highly skilled or professional jobs than the Amber Hill adults... 65 percent of the Phoenix Park adults were in jobs more professional than their parents, compared with 23 percent of Amber Hill adults. In fact, 52 percent of Amber Hill adults were in less professional jobs than their parents, compared with only 15 percent of the Phoenix Park graduates...
Students at the school that that taught problem solving rather than memorization continued math education in college because they enjoyed it, leading 65% of them to get better jobs than their parents at only 24 years old. On the other hand, the majority of students at the traditional high school avoided math in college, and chose "less professional" jobs than their parents at 24. Over their lifespans, I can only imagine the disparity increased...
> Which statement would give you more pleasure? "Because of good teaching, my child scored 79% on her last math test," or "Because of good teaching, my child has a much better job and leads a far more interesting and rewarding life than me."
The author mentions that even the best students at traditional schools had trouble on tests since they practiced the skill-of-the-day on assignments, leading me to believe they would show remarkable improvement on standardized testing in a new curriculum. Maybe the author believes that testing students after-the-fact is not as good a use of taxpayer money as providing resources and education for teachers to change the curriculum would be.
For context here: Phoenix Park was predominantly working class, while Amber Hill was predominantly "affluent".
In this context, the results sound a lot like reversion the mean.
In this context, the results sound a lot like reversion the mean.
" In Math You Have to Remember, In Other Subjects You Can Think About It"
Most math geniuses I know (including however much of a genius I am) know the exact opposite! Other subjects are memorizing facts; you can noodle through math problems using logic. In fact, much of our success in other subjects is in weaseling through multiple-guess tests using mathematical logic!
Most math geniuses I know (including however much of a genius I am) know the exact opposite! Other subjects are memorizing facts; you can noodle through math problems using logic. In fact, much of our success in other subjects is in weaseling through multiple-guess tests using mathematical logic!
You should read the article instead of responding to the title, which is in fact a quote by a student about the current state of education in the US.
I was helping my 9th grade step brother with his math homework, and I've noticed his math classes are no longer divided into subjects they way they where when I was there 10 years ago (algebra, geometry, algebra II, trig/precalc, calculus, discrete math)
His math book jumped around so much, that one week he was working on probability, the next basic geometry, and the next simple factoring.
It looked exactly like someone wrote a whirlwind study guide that covered just enough to pass some very specific standardized test.
His math book jumped around so much, that one week he was working on probability, the next basic geometry, and the next simple factoring.
It looked exactly like someone wrote a whirlwind study guide that covered just enough to pass some very specific standardized test.
Personally, I would have preferred a breadth-first approach to mathematics when I was in school. All of those concepts are related, but it's very difficult to make those connections when you are "only learning [algebra|geometry|etc]" in isolation.
Edit: related http://steve-yegge.blogspot.com/2006/03/math-for-programmers...
Edit: related http://steve-yegge.blogspot.com/2006/03/math-for-programmers...
This definitely isn't some kind of holistic approach to math. It's basically just here memorize this probability forumla, and the algorithm for multiplying two brackets together, because that's what you're getting tested on at the end of the year.
I could understand if it was something like showing how bracket multplication works by adding in an example from geometry like this http://math1sfun.wordpress.com/category/algebra/expanding-br...
But it's nothing like that at all.
I could understand if it was something like showing how bracket multplication works by adding in an example from geometry like this http://math1sfun.wordpress.com/category/algebra/expanding-br...
But it's nothing like that at all.
It is still split up that way, but I think that is for more advanced students. For students not focusing on math, they may just make a text that generalizes on many topics without focusing on a specific subject.
>It is still split up that way, but I think that is for more advanced students.
He's in honors math, i.e., the most advanced math available for his grade level.
He's in honors math, i.e., the most advanced math available for his grade level.
Math is about thinking abstractly. To do it at the level required by modern science, data analysis and engineering you need to be able to focus on the abstract symbols, equations and rules that govern them without relying on an intuition for underling objects. For example, no one has valid intuition for fluid turbulence, n dimensional manifold theory or complicated probability distributions, so great leeps in understanding these are made by people who have an intuition for how equations behave and rigorously show that it is valid. Real world applications are often only found after the fact.
I think that they key to a good math education is not just showing students real world application but teaching them to find beauty and pleasure in abstract symbolic reasoning and the rigors of proof.