5 ms·

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

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.

wortiz 17y ago
To answer your last question first I think traveling problems are taught in math because speed and time are easy for students to relate to, the same way that finding the height of a flagpole by its shadow problems are easy for students to visualize and relate to. As for average speed that is one of those concepts that really isn't that relatable and should probably require an introduction beforehand. As for the goal of a math class being that students should come out understanding math I completely agree but the beauty of math is that it is applicable everywhere and the goal of teaching a math class should be that students understand mathematics and where to apply it. I couldn't imagine being taught vector calculus without some mention of applications and examples, my professor catered more towards mathematics majors with an abundance of Tensor notation and proofs but he never failed to understand that many in his class learned well by physical and other practical examples such as magnetism or fluids and so on. Although I failed to watch your video I believe there is most likely a counterargument to word problems and what effect they have on students because if students really are solving word problems with the math they are learning the issue isn't that they don't understand what is going on but rather how to apply it to an actual math problem. Either that or the word problems do not incorporate well enough with the math that a student can grasp it quickly without knowing underlying ideas.
- yequalsx 17y ago
  The experience of the professor at Harvard in the video contradicts your last paragraph. They know how to apply the math but they don't understand the problem. And the traveling problem I gave underscores this point. It's a very simple problem and almost no one gets it correct.
  - wortiz 17y ago
    Then perhaps you and him are assigning the wrong word problems or not giving proper introductions to said word problems.
    - yequalsx 17y ago
      Perhaps you should watch the video and then you would realize that your characterization of this professor and me is wrong.
    - argv_empty 17y ago
      If all students are taught to do is "memorization of problem types" (to borrow GP's term), then they will only be able to handle actual problems that match one of the necessarily finite set of types of problems they have memorized. A different selection of problem types to memorize will not provide general coverage -- that requires learning to construct a solution to a new type of problem.
      - nitrogen 17y ago
        Nowhere in the thread has it been demonstrated that it would be impossible to design a word problem that does require the construction of a solution. One of the problems I had with my dif. eq. class was that the grad student teaching the class just focused on "tank problems" and other "x problems" sets. So I see why story problems alone are not helpful. But, if they're judiciously used to encourage the understanding of abstract mathematical concepts, I think they are invaluable.
adamtj 17y ago
His argument is that current HS math doesn't provide enough distractions. Students don't learn to focus on the problem at hand, because the problem is already in sharp focus. All the variables are provided. The word problem is a slightly jumbled description of the formula to use. All you have to do is look up the formula, then plug in the variables. Your students can't solve the travelling problem for the very reasons he's talking about. That problem is completely unlike anything they've ever solved in high school, because it doesn't give them the numbers to average. It's a two step problem where they need to use the given numbers in one formula, to get the resulting numbers for the second. HS students don't learn to solve problems: they learn to match paragraphs to formulas, then plug and chug. Also, don't be such a purist. It's not useful. There is a great deal of overlap between math and physics. Physicists often invent new math to understand physics, and old math is often found to describe some newly discovered physical phenomenon.
- yequalsx 17y ago
  You wrote: "HS students don't learn to solve problems: they learn to match paragraphs to formulas, then plug and chug." I agree completely and this is why I advocate for an end to the focus on so called practical word problems in high school in algebra classes. It should be in physics where one learns about travelling problems not in algebra. The focus on so called practical word problems is not accomplishing the goal and it's distracting from the learning of mathematics.
  - dhimes 17y ago
    I taught physics in a community college as a full-time prof for 10 years, and had the occasion to teach math there a few times as well. After that stint, I started work with an ed software company developing math content for algebra students in the cc. In my experience, the "word problems" can be used to motivate the students as to the interesting part of what they are studying. My focus wasn't on making them learn how to reason through complicated applications, it was on getting them to master the skills. So in that sense I agree with you. However, building a story around it (how do you determine which cell phone plan to buy?) helps keep their attention and shows them that they- yes THEY (the "bottom of the barrel" as was noted above- but I'll clarify this below)-- can apply math to their lives and live more intelligently. And getting students to that stage is my life's mission. I think where math goes wrong is displayed beautifully in the rational expressions units (I didn't go to the original link, so forgive me if I'm entirely redundant here). When I was building the math content for the ed s/w company I was struck that nowhere in the chapter where students learn to simplify fractions involving a polynomial numerator and a polynomial denominator (tough stuff when you are first learning it) nowhere in the textbooks did they tell the student WHY. Why do all this hard work? What is the answer that we seek from all this? [rant] About the idea of "stupid teachers teaching stupid students" in community college: Anyone who truly speaks from experience knows better. In a cosmopolitan setting you will have downright brilliant students in your classes- especially in your night classes. I've had many ivy-leaguers in my classes, top accountants for Exxon-Mobile and other high-power operators, sometimes from government agencies (I taught in Alexandria Virginia, about a mile from the Pentagon, where Mrs. Biden currently teaches). A common reason for them to be there? They've burned out in their careers and want to do something that "makes a difference." Going to med or vet school was common. They've got the degree, they've got the grades, they just need a couple of classes that they didn't take as undergrads, they're too smart to spend the money on a school that won't help them any more than a little ol' cc will. Keeping their job for an extra year and taking night classes is a smart move, so they come to the local cc. [/rant]

wortiz 17y ago

To answer your last question first I think traveling problems are taught in math because speed and time are easy for students to relate to, the same way that finding the height of a flagpole by its shadow problems are easy for students to visualize and relate to. As for average speed that is one of those concepts that really isn't that relatable and should probably require an introduction beforehand. As for the goal of a math class being that students should come out understanding math I completely agree but the beauty of math is that it is applicable everywhere and the goal of teaching a math class should be that students understand mathematics and where to apply it. I couldn't imagine being taught vector calculus without some mention of applications and examples, my professor catered more towards mathematics majors with an abundance of Tensor notation and proofs but he never failed to understand that many in his class learned well by physical and other practical examples such as magnetism or fluids and so on. Although I failed to watch your video I believe there is most likely a counterargument to word problems and what effect they have on students because if students really are solving word problems with the math they are learning the issue isn't that they don't understand what is going on but rather how to apply it to an actual math problem. Either that or the word problems do not incorporate well enough with the math that a student can grasp it quickly without knowing underlying ideas.

yequalsx 17y ago
The experience of the professor at Harvard in the video contradicts your last paragraph. They know how to apply the math but they don't understand the problem. And the traveling problem I gave underscores this point. It's a very simple problem and almost no one gets it correct.
- wortiz 17y ago
  Then perhaps you and him are assigning the wrong word problems or not giving proper introductions to said word problems.
  - yequalsx 17y ago
    Perhaps you should watch the video and then you would realize that your characterization of this professor and me is wrong.
  - argv_empty 17y ago
    If all students are taught to do is "memorization of problem types" (to borrow GP's term), then they will only be able to handle actual problems that match one of the necessarily finite set of types of problems they have memorized. A different selection of problem types to memorize will not provide general coverage -- that requires learning to construct a solution to a new type of problem.
    - nitrogen 17y ago
      Nowhere in the thread has it been demonstrated that it would be impossible to design a word problem that does require the construction of a solution. One of the problems I had with my dif. eq. class was that the grad student teaching the class just focused on "tank problems" and other "x problems" sets. So I see why story problems alone are not helpful. But, if they're judiciously used to encourage the understanding of abstract mathematical concepts, I think they are invaluable.

adamtj 17y ago

His argument is that current HS math doesn't provide enough distractions. Students don't learn to focus on the problem at hand, because the problem is already in sharp focus. All the variables are provided. The word problem is a slightly jumbled description of the formula to use. All you have to do is look up the formula, then plug in the variables. Your students can't solve the travelling problem for the very reasons he's talking about. That problem is completely unlike anything they've ever solved in high school, because it doesn't give them the numbers to average. It's a two step problem where they need to use the given numbers in one formula, to get the resulting numbers for the second. HS students don't learn to solve problems: they learn to match paragraphs to formulas, then plug and chug. Also, don't be such a purist. It's not useful. There is a great deal of overlap between math and physics. Physicists often invent new math to understand physics, and old math is often found to describe some newly discovered physical phenomenon.

yequalsx 17y ago
You wrote: "HS students don't learn to solve problems: they learn to match paragraphs to formulas, then plug and chug." I agree completely and this is why I advocate for an end to the focus on so called practical word problems in high school in algebra classes. It should be in physics where one learns about travelling problems not in algebra. The focus on so called practical word problems is not accomplishing the goal and it's distracting from the learning of mathematics.
- dhimes 17y ago
  I taught physics in a community college as a full-time prof for 10 years, and had the occasion to teach math there a few times as well. After that stint, I started work with an ed software company developing math content for algebra students in the cc. In my experience, the "word problems" can be used to motivate the students as to the interesting part of what they are studying. My focus wasn't on making them learn how to reason through complicated applications, it was on getting them to master the skills. So in that sense I agree with you. However, building a story around it (how do you determine which cell phone plan to buy?) helps keep their attention and shows them that they- yes THEY (the "bottom of the barrel" as was noted above- but I'll clarify this below)-- can apply math to their lives and live more intelligently. And getting students to that stage is my life's mission. I think where math goes wrong is displayed beautifully in the rational expressions units (I didn't go to the original link, so forgive me if I'm entirely redundant here). When I was building the math content for the ed s/w company I was struck that nowhere in the chapter where students learn to simplify fractions involving a polynomial numerator and a polynomial denominator (tough stuff when you are first learning it) nowhere in the textbooks did they tell the student WHY. Why do all this hard work? What is the answer that we seek from all this? [rant] About the idea of "stupid teachers teaching stupid students" in community college: Anyone who truly speaks from experience knows better. In a cosmopolitan setting you will have downright brilliant students in your classes- especially in your night classes. I've had many ivy-leaguers in my classes, top accountants for Exxon-Mobile and other high-power operators, sometimes from government agencies (I taught in Alexandria Virginia, about a mile from the Pentagon, where Mrs. Biden currently teaches). A common reason for them to be there? They've burned out in their careers and want to do something that "makes a difference." Going to med or vet school was common. They've got the degree, they've got the grades, they just need a couple of classes that they didn't take as undergrads, they're too smart to spend the money on a school that won't help them any more than a little ol' cc will. Keeping their job for an extra year and taking night classes is a smart move, so they come to the local cc. [/rant]