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I'm a teacher of mathematics at a community college. It appears to me that the crux of the problem is that people, including the guy in the video, confuse prob

I'm a teacher of mathematics at a community college. It appears to me that the crux of the problem is that people, including the guy in the video, confuse problem solving with mathematics. The utility of mathematics comes in the remarkable fact that a great deal of phenomena can be adequately modeled mathematically. The focus of an algebra class ought to be in learning the language of algebra. That is, in manipulating numbers in the abstract. The application problems ought to be saved for physics, biology, economics, etc. The result of an emphasis on so called real world problems in high school mathematics is a generation of students who are incapable of correctly manipulating algebraic expressions and equations. I recently gave my college algebra class an equation. It was a simple equation and all of them could solve it. I then asked them for an example of an equation that had no solution. Not a single person could provide an answer. They can solve the word problems in the textbook but don't have the slightest clue about what these mathematical concepts actually mean. Perhaps it isn't important that one need to manipulate algebraic objects. I won't argue with this. But let's not call solving word problems mathematics. If you want to learn mathematics then grinding through the minutia and having the patience to understand the symbols is necessary. You can't get around this. If the goal is to solve word problems then....go ahead and change things.

anuleczka 17y ago
In a world where Wolfram Alpha can do all the algebraic calculations for you, isn't learning to solve problems more relevant? I'd say that problem solving is the core of mathematics. Am I wrong?
- flatline 17y ago
  The conceptual understanding is certainly essential, but if you want to use a computer to do symbolic/algebraic manipulation, how could you be sure to understand the results or even properly phrase the question without a thorough understanding of the symbols and the rules for their manipulation? The best I think that something like Alpha could do is to help elucidate the underlying principles or give you a quick answer for guesswork or estimation, like a calculator.
  - jey 17y ago
    You contradict yourself. As you said, one does not need to be skilled in combinatorial search, er I mean, algebraic manipulations to be able to solve problems; one only needs a solid conceptual grounding and understand the symbols and the rules for their manipulation. I wouldn't be surprised if most students who pass college calculus courses with good grades are entirely unable to recognize the situations where calculus is applicable.
- yequalsx 17y ago
  Yeah, Wolframalpha is a disruptive technology. Is it still relevant to learn algebraic manipulations? I don't know. There are compelling arguments for both points of view. When you say problem solving do you really mean so called practical applications? If so then I disagree. If you mean solving problems to understand the essence of mathematical ideas then I agree. For instance, linear equations are the easiest equations to solve. All of them can be manipulated to yield the solution (or be reduced to showing all numbers are solutions or have no solutions). Linear equations are really 1st degree polynomials. What about second degree polynomials? After a clever bit of insight and some work we get that all 2nd degree polynomials can be solved. The quadratic equation is a solution to a problem in mathematics. It's useful as it leads to a greater understanding of mathematics. I believe that this is what mathematics courses ought to focus on. Not on problems that involve throwing a ball in the air and predicting how long it takes to hit the ground.
  - jimbokun 17y ago
    "Yeah, Wolframalpha is a disruptive technology. Is it still relevant to learn algebraic manipulations?" We've had calculators for decades now, and my kids are still learning how to do arithmetic.
    - yequalsx 17y ago
      But is it worthwhile or important?
      - jimbokun 17y ago
        I think it is. If you can't perform the algorithms yourself, I don't think you really understand what the symbols on the machine really mean. When I worked at a bank, I remember complaints about an employee whose job involved running certain calculations in a spreadsheet. Sometimes the numbers would be off orders of magnitude due to some error somewhere, but the employee would just carry on as normal. If you have some clue as to what the calculations are supposed to be doing, there should be some alarms going off if the numbers are not at least reasonable. Or what if you just wrote a program that includes some arithmetic calculations, and need to verify that you are getting the right answers for your test data? I remember another time at that same bank, where a developer did not know that multiplication had precedence over addition, which led to errors in a financial calculation. I'm sure you could easily think up many more examples like this.
- deleted 17y ago
  [deleted]
- zackattack 17y ago
  Let me share a frustrating experience I recently had. I hired some dude off Craigslist to install my TV wall mount because I don't have the proper tools. I tried to explain where I wanted the TV: I want it to be centered in the wall, but then moved a few inches to the right. He simply couldn't understand what I was saying. It blew my mind, I had to just measure it out myself and mark the spot. If he had a solid of understanding of basic high school algebra, he would have been able to formulate the problem in his head. Done.
wortiz 17y ago
I'm going to chime in from experience and say that your college algebra students are the bottom of the barrel in regards to math students. With that respects there can be many reasons why they are there: they didn't care about math, they never really understood math, and/or they never really tried to get math. If you are judging the ability of finishing word problems in the textbook, I'm assuming for homework assignments, and they really don't understand the actual math I think your problem is probably that most are using a solutions manual or spending what would seem too much time solving the homework problems. As for solving word problems or practical problems I believe that this is exactly what math should be for especially at the college algebra level. To be serious most students entering college who want or are going to pursue a degree in mathematics are going to start off closer to the calculus level and maybe trigonometry depending on underlying circumstances. What people in college algebra are going for is a degree where math has to be practical and the more diverse applications they see the better. To give a simple example take logarithms: if you were to teach logarithms on the sole basis of algebraic symbols and expressions when it comes time that these students need to apply logarithms to real world problems (like say compound interest) there is going to be a good chance that they didn't really comprehend logs when they were taught them and won't be able to use them properly. Which brings me to another point for word problems: word problems exist because they make students think and relate math to the real world it is fundamental to learning for many students to apply mathematical concepts in a multitude of ways in order to actual grasp a concept and one of the easiest ways to do this is to place that concept into word problems.
- yequalsx 17y ago
  I strongly disagree with the notion that college algebra ought to be the place where one learns practical applications of mathematics. A valuable skill for any educated person is the ability to plod through minutia and make sense of it. Being able to be focused on the point at hand and not let distractions cloud judgment is a valuable skill. If one really understands the mechanics of logs then I think they will not freak out when presented with a formula involving logs. I'm all in favor of applications but isn't this the job of other subjects? The applications should be taught in the subjects that apply it. The goal of a mathematics class ought to be to understand mathematics. This lecture, http://www.youtube.com/watch?v=WwslBPj8GgI http://www.youtube.com/watch?v=WwslBPj8GgI is long but well worth watching. He provides strong evidence that students learn to solve word problems but haven't the slightest idea of what is really going on. They are memorizing steps. This happens at Harvard and the community college. I've given my students the standard word problems on travelling problems. They get the problems right. Then I ask them this simple problem and they universally get is wrong: You drive 1 mile at 40 mph and then the next mile drive 60 mph. What is your average speed? (Almost everyone says 50 mph.) We aren't teaching thinking skills. We are teaching memorization of problem types. Almost none of them is actually practical, correct, or useful. But why are we teaching travelling problems in a math class? I think it should be taught in physics.
adamtj 17y ago
During training, soccer players often run around the field in a big group, but you'll never see them do that during a game. Running laps around the field isn't the goal, but it does help them achieve their goals. His basic argument is that the point of high school math education is to teach kids how to reason and understand the world quantitatively. I agree. The teaching of algebra to high school students is secondary to the real goal. Without a good education, most people are sloppy thinkers -- and they vote more frequently than they solve equations.
- yequalsx 17y ago
  Teaching so called practical word problems is not accomplishing this goal.
olliesaunders 17y ago
You can give yours students an incredibly useful skill that has inordinate practical application, exciting their minds with possibilities, intrigue, and a sense of empowerment. Or you can engage them in rote mechanics, fail to impart the usefulness of mathematics, and send them all to sleep.
- yequalsx 17y ago
  Great generalization and oversimplification. How about this oversimplification? Or you can teach a generation of students that if isn't 'practical' then it shouldn't be done and certainly isn't worth any effort or struggle. Hopefully then, we can eliminate poetry, music, art, literature, and p.e. We have practical problems to solve!
  - olliesaunders 17y ago
    I consider engaging and consuming forms of art akin to play. At first sight to spend lots of time playing might be seen as a waste of time but the fact that a child prevented from playing would fail to develop properly says something about its purpose. Basically I don't agree that those subjects are purposeless.
    - yequalsx 17y ago
      That's great that you consider this. Many do not consider poetry, music, art classes worthwhile because they consider those classes impractical. I don't agree that learning how to abstractly apply rules and manipulate objects is useless. It's particularly important in programming, for instance.
      - nitrogen 17y ago
        I think the other poster was suggesting that students will perceive abstract symbolic manipulation as useless unless they're taught why it isn't useless, probably via application problems. Considering that the majority of students will never have a genuine interest in math for its own sake, I think that textbooks and curricula designed for public education should include high doses of application problems, with enough theoretical problems to interest the true mathematicians and verify the students' understanding of the abstract concepts.
dustingetz 17y ago
addressing this statement in a vaccuum: "The focus of an algebra class ought to be in learning the language of algebra" i think learning the relevance of its applications goes hand-in-hand and is necessary for a deeper understanding.
glyn 17y ago
While I'd agree that students often approach word problems by basically memorizing mappings between a problem class and a small set of equations, I'm not sure that merely teaching them the mechanics of mathematics in the abstract is sufficient for imparting a thorough, practical understanding of the subject. My experience with math-related courses was that I had the greatest difficulty generalizing the word-problem aspects into being able to reason about novel problems. A lot of the lectures and texts kind of paid lip service to the idea of problem solving in a general manner, but I think there's a lot of room for improvement in terms of capturing and teaching the underlying requisite types of mental activity, in a way that's explicitly procedural.
- yequalsx 17y ago
  I agree with you. The problem I've encountered is that students who lack intellectual curiosity aren't motivated by the practical applications found in algebra textbooks. They aren't motivated by any sort of problem. So there is a reduction to the lowest denominator. That is, we give word problems that are reducible to mappings to a small set of equations.
stan_rogers 17y ago
I wrote a bit about this a few years ago, before my stroke: http://stanrogers.blogspot.com/2005/03/girls-math-stuff.html http://stanrogers.blogspot.com/2005/03/girls-math-stuff.html Warning: the site in general is a personal blog, so don't expect to see much else of interest to anyone.

anuleczka 17y ago

In a world where Wolfram Alpha can do all the algebraic calculations for you, isn't learning to solve problems more relevant? I'd say that problem solving is the core of mathematics. Am I wrong?

flatline 17y ago
The conceptual understanding is certainly essential, but if you want to use a computer to do symbolic/algebraic manipulation, how could you be sure to understand the results or even properly phrase the question without a thorough understanding of the symbols and the rules for their manipulation? The best I think that something like Alpha could do is to help elucidate the underlying principles or give you a quick answer for guesswork or estimation, like a calculator.
- jey 17y ago
  You contradict yourself. As you said, one does not need to be skilled in combinatorial search, er I mean, algebraic manipulations to be able to solve problems; one only needs a solid conceptual grounding and understand the symbols and the rules for their manipulation. I wouldn't be surprised if most students who pass college calculus courses with good grades are entirely unable to recognize the situations where calculus is applicable.
yequalsx 17y ago
Yeah, Wolframalpha is a disruptive technology. Is it still relevant to learn algebraic manipulations? I don't know. There are compelling arguments for both points of view. When you say problem solving do you really mean so called practical applications? If so then I disagree. If you mean solving problems to understand the essence of mathematical ideas then I agree. For instance, linear equations are the easiest equations to solve. All of them can be manipulated to yield the solution (or be reduced to showing all numbers are solutions or have no solutions). Linear equations are really 1st degree polynomials. What about second degree polynomials? After a clever bit of insight and some work we get that all 2nd degree polynomials can be solved. The quadratic equation is a solution to a problem in mathematics. It's useful as it leads to a greater understanding of mathematics. I believe that this is what mathematics courses ought to focus on. Not on problems that involve throwing a ball in the air and predicting how long it takes to hit the ground.
- jimbokun 17y ago
  "Yeah, Wolframalpha is a disruptive technology. Is it still relevant to learn algebraic manipulations?" We've had calculators for decades now, and my kids are still learning how to do arithmetic.
  - yequalsx 17y ago
    But is it worthwhile or important?
    - jimbokun 17y ago
      I think it is. If you can't perform the algorithms yourself, I don't think you really understand what the symbols on the machine really mean. When I worked at a bank, I remember complaints about an employee whose job involved running certain calculations in a spreadsheet. Sometimes the numbers would be off orders of magnitude due to some error somewhere, but the employee would just carry on as normal. If you have some clue as to what the calculations are supposed to be doing, there should be some alarms going off if the numbers are not at least reasonable. Or what if you just wrote a program that includes some arithmetic calculations, and need to verify that you are getting the right answers for your test data? I remember another time at that same bank, where a developer did not know that multiplication had precedence over addition, which led to errors in a financial calculation. I'm sure you could easily think up many more examples like this.
deleted 17y ago
[deleted]
zackattack 17y ago
Let me share a frustrating experience I recently had. I hired some dude off Craigslist to install my TV wall mount because I don't have the proper tools. I tried to explain where I wanted the TV: I want it to be centered in the wall, but then moved a few inches to the right. He simply couldn't understand what I was saying. It blew my mind, I had to just measure it out myself and mark the spot. If he had a solid of understanding of basic high school algebra, he would have been able to formulate the problem in his head. Done.

wortiz 17y ago

I'm going to chime in from experience and say that your college algebra students are the bottom of the barrel in regards to math students. With that respects there can be many reasons why they are there: they didn't care about math, they never really understood math, and/or they never really tried to get math. If you are judging the ability of finishing word problems in the textbook, I'm assuming for homework assignments, and they really don't understand the actual math I think your problem is probably that most are using a solutions manual or spending what would seem too much time solving the homework problems. As for solving word problems or practical problems I believe that this is exactly what math should be for especially at the college algebra level. To be serious most students entering college who want or are going to pursue a degree in mathematics are going to start off closer to the calculus level and maybe trigonometry depending on underlying circumstances. What people in college algebra are going for is a degree where math has to be practical and the more diverse applications they see the better. To give a simple example take logarithms: if you were to teach logarithms on the sole basis of algebraic symbols and expressions when it comes time that these students need to apply logarithms to real world problems (like say compound interest) there is going to be a good chance that they didn't really comprehend logs when they were taught them and won't be able to use them properly. Which brings me to another point for word problems: word problems exist because they make students think and relate math to the real world it is fundamental to learning for many students to apply mathematical concepts in a multitude of ways in order to actual grasp a concept and one of the easiest ways to do this is to place that concept into word problems.

yequalsx 17y ago
I strongly disagree with the notion that college algebra ought to be the place where one learns practical applications of mathematics. A valuable skill for any educated person is the ability to plod through minutia and make sense of it. Being able to be focused on the point at hand and not let distractions cloud judgment is a valuable skill. If one really understands the mechanics of logs then I think they will not freak out when presented with a formula involving logs. I'm all in favor of applications but isn't this the job of other subjects? The applications should be taught in the subjects that apply it. The goal of a mathematics class ought to be to understand mathematics. This lecture, http://www.youtube.com/watch?v=WwslBPj8GgI http://www.youtube.com/watch?v=WwslBPj8GgI is long but well worth watching. He provides strong evidence that students learn to solve word problems but haven't the slightest idea of what is really going on. They are memorizing steps. This happens at Harvard and the community college. I've given my students the standard word problems on travelling problems. They get the problems right. Then I ask them this simple problem and they universally get is wrong: You drive 1 mile at 40 mph and then the next mile drive 60 mph. What is your average speed? (Almost everyone says 50 mph.) We aren't teaching thinking skills. We are teaching memorization of problem types. Almost none of them is actually practical, correct, or useful. But why are we teaching travelling problems in a math class? I think it should be taught in physics.

adamtj 17y ago

During training, soccer players often run around the field in a big group, but you'll never see them do that during a game. Running laps around the field isn't the goal, but it does help them achieve their goals. His basic argument is that the point of high school math education is to teach kids how to reason and understand the world quantitatively. I agree. The teaching of algebra to high school students is secondary to the real goal. Without a good education, most people are sloppy thinkers -- and they vote more frequently than they solve equations.

yequalsx 17y ago
Teaching so called practical word problems is not accomplishing this goal.

olliesaunders 17y ago

You can give yours students an incredibly useful skill that has inordinate practical application, exciting their minds with possibilities, intrigue, and a sense of empowerment. Or you can engage them in rote mechanics, fail to impart the usefulness of mathematics, and send them all to sleep.

yequalsx 17y ago
Great generalization and oversimplification. How about this oversimplification? Or you can teach a generation of students that if isn't 'practical' then it shouldn't be done and certainly isn't worth any effort or struggle. Hopefully then, we can eliminate poetry, music, art, literature, and p.e. We have practical problems to solve!
- olliesaunders 17y ago
  I consider engaging and consuming forms of art akin to play. At first sight to spend lots of time playing might be seen as a waste of time but the fact that a child prevented from playing would fail to develop properly says something about its purpose. Basically I don't agree that those subjects are purposeless.
  - yequalsx 17y ago
    That's great that you consider this. Many do not consider poetry, music, art classes worthwhile because they consider those classes impractical. I don't agree that learning how to abstractly apply rules and manipulate objects is useless. It's particularly important in programming, for instance.
    - nitrogen 17y ago
      I think the other poster was suggesting that students will perceive abstract symbolic manipulation as useless unless they're taught why it isn't useless, probably via application problems. Considering that the majority of students will never have a genuine interest in math for its own sake, I think that textbooks and curricula designed for public education should include high doses of application problems, with enough theoretical problems to interest the true mathematicians and verify the students' understanding of the abstract concepts.

dustingetz 17y ago

addressing this statement in a vaccuum: "The focus of an algebra class ought to be in learning the language of algebra" i think learning the relevance of its applications goes hand-in-hand and is necessary for a deeper understanding.

glyn 17y ago

While I'd agree that students often approach word problems by basically memorizing mappings between a problem class and a small set of equations, I'm not sure that merely teaching them the mechanics of mathematics in the abstract is sufficient for imparting a thorough, practical understanding of the subject. My experience with math-related courses was that I had the greatest difficulty generalizing the word-problem aspects into being able to reason about novel problems. A lot of the lectures and texts kind of paid lip service to the idea of problem solving in a general manner, but I think there's a lot of room for improvement in terms of capturing and teaching the underlying requisite types of mental activity, in a way that's explicitly procedural.

yequalsx 17y ago
I agree with you. The problem I've encountered is that students who lack intellectual curiosity aren't motivated by the practical applications found in algebra textbooks. They aren't motivated by any sort of problem. So there is a reduction to the lowest denominator. That is, we give word problems that are reducible to mappings to a small set of equations.

stan_rogers 17y ago

I wrote a bit about this a few years ago, before my stroke: http://stanrogers.blogspot.com/2005/03/girls-math-stuff.html http://stanrogers.blogspot.com/2005/03/girls-math-stuff.html Warning: the site in general is a personal blog, so don't expect to see much else of interest to anyone.