6 ms·

I think the simplest approach is to use the cube root function. This is the simplest approach because it solves, over the reals, any equation of the form x^3

I think the simplest approach is to use the cube root function. This is the simplest approach because it solves, over the reals, any equation of the form x^3 = real number That's the simplest solution. It works in every case. To me the answer is not important. The methodology is important. Giving a counter example is very much a different type of problem. Just about any method is valid in that type of problem. Passing a class should mean more than I got a lot of answers correct. It should mean an understanding of the material. A college algebra student who solves x^3=27 in the aforementioned manner is lacking a fundamental understanding of the material. Now a third grader who reasons thusly, well that is impressive. The goal is not the right answer. It a demonstration of understanding and abilit appropriate to the level of the course.

proactive 10y ago
You and I think that the cube root function is the simplest approach. The student doesn't necessarily agree. And I disagree with you that the most general approach is necessarily the simplest. > Just about any method is valid in that type of problem. Why is that not true for other types of problems? > Passing a class should mean more than I got a lot of answers correct. I agree. But you shouldn't penalise the student if the exam question is poorly framed (and we all make such mistakes). Just take a note for later and don't make the mistake again.
- yequalsx 10y ago
  Let's take a calc 2 example. I ask students to integrate ln(x). I want to know if they can do integration by parts when one function is 1. Some of them can memorize the answer and just write it down. I don't give them credit for this. I'm giving an easy problem because I just want to know if they know how to do parts with 1 as one of the functions. I don't want to load the test with hard problems so that I can eliminate any possibility of memorization at play. It's interesting reading all the replies I've gotten. It's nice to see other peoples' perspectives. Including yours. As you stated I would not give x^3=27 as a problem in college algebra. It's a fine line and I suspect that we mostly agree except on one part. As a grader I've given full credit for the wrong answer and no credit for the right answer.
  - gizmo686 10y ago
    So, your proposed solution to integral(ln(x)) involves first noticing that d/dx (1/x) = ln(x), then proceeding to use v=1/x to integrate by parts? I think I am misunderstanding something, because as far as I can tell, integration by parts where one of the parts is 1 is literally useless.
    - yequalsx 10y ago
      u = ln(x) and dv = 1 du = 1/x and v = x integral ln(x) = x ln(x) - integral 1/x times x integral ln(x) = x ln(x) - x + C I could give integral arctan(x) but with the advent of computer algebra systems I'm mostly interested in them knowing the basic examples and to not burden them on a test with something more complicated. EDIT: The derivative of 1/x is not ln(x) as you stated. You got it backwards and my guess is that is the source of your confusion.
webmaven 10y ago
> Passing a class should mean more than I got a lot of answers correct. It should mean an understanding of the material. I agree, but testing for understanding (as opposed to Socratically probing for it) is more time consuming, complex, and difficult than just testing for correct answers. To stay within a given test workload, students would have to take far fewer tests. Obviously not the direction the educational system is trending these days. Which is a shame.
- Ntrails 10y ago
  >I agree, but testing for understanding is more time consuming, complex, and difficult than just testing for correct answers. You get 1 mark for the right answer and 2 marks for working. Problem solved. The question doesn't have to change at all. This is how I remember mark schemes working in the general case
  - webmaven 10y ago
    > You get 1 mark for the right answer and 2 marks for working. I'm not sure I get what you mean by "working". Can you expand that a bit?
BeetleB 10y ago
As someone who has spent time in the math world, I sympathize with proactive's response. Once you get to a "high enough" level, the lessons you learned earlier have to be unlearned. Not just true in mathematics but in engineering as well. I was taking an engineering course in my sophomore year where I had a system of equations in 3-4 variables. I spent forever trying to solve it (analytically), and failed. So did most of the class. The next lecture, the professor showed us how to do it. A mixture of plots, etc reduced the solution space and the rest was trivial. He also said "You could just use the solver in your calculator/MATLAB". I wasn't satisfied with his answer. It felt like cheating. I didn't learn the cool way to do things. But in the real world, if you can get the solution this way, it's perfectly valid. As long as you can confirm that you found a/the solution (trivial to do). With the x^3=27 answer, it is the onus of the instructor to specify explicitly that "guessing is not allowed". Why? Because as others have mentioned, it is totally appropriate in mathematical circles to guess a solution. Much work in mathematics is done that way. In various classes (mathematics/engineering/physics), I've both utilized non-standard ways to solve problems on tests, and have seen it done by students on tests I grade. This is to be encouraged. Especially because this is what mathematicians/physicists love to do in their real work. If your goal is to ensure they understand cube roots, either make a problem that is hard to guess (e.g. x^3 = 24), or be explicit about it. Even with x^3 = 24, if they use Newton's method, that should be graded correct.
- yequalsx 10y ago
  Professional mathematicians don't guess answers to theorems. They sometimes guess counterexamples. But no one guesses answers to a theorem. I've seen, "I notice that A is a solution to this equation does anyone know a method for formally solving it?" Guessing is not a method of solving. It is a method of finding counterexamples. Two different types of problems. I accept any mathematically valid method of solving a problem. Mathematically method means, method that works even if I trivially change it by using different numbers.
  - BeetleB 10y ago
    >Professional mathematicians don't guess answers to theorems. Believe it or not, proving theorems is not the goal of many mathematicians. I'm simplifying a bit, but read Freeman Dyson's essay on Birds and Frogs. Essentially "problem solvers" vs "theory builders". While problem solvers often do end up proving theorems, it is not their main goal. If they can "guess" a solution, they are done. It is publishable. Go to the field of combinatorics, and you'll find it is full of guesswork. >I accept any mathematically valid method of solving a problem. Mathematically method means, method that works even if I trivially change it by using different numbers. Sorry, but many mathematicians disagree with you. Solving a problem is finding a solution (provided you have a means to verify correctness). It doesn't matter if you merely guessed it.

proactive 10y ago

You and I think that the cube root function is the simplest approach. The student doesn't necessarily agree. And I disagree with you that the most general approach is necessarily the simplest. > Just about any method is valid in that type of problem. Why is that not true for other types of problems? > Passing a class should mean more than I got a lot of answers correct. I agree. But you shouldn't penalise the student if the exam question is poorly framed (and we all make such mistakes). Just take a note for later and don't make the mistake again.

yequalsx 10y ago
Let's take a calc 2 example. I ask students to integrate ln(x). I want to know if they can do integration by parts when one function is 1. Some of them can memorize the answer and just write it down. I don't give them credit for this. I'm giving an easy problem because I just want to know if they know how to do parts with 1 as one of the functions. I don't want to load the test with hard problems so that I can eliminate any possibility of memorization at play. It's interesting reading all the replies I've gotten. It's nice to see other peoples' perspectives. Including yours. As you stated I would not give x^3=27 as a problem in college algebra. It's a fine line and I suspect that we mostly agree except on one part. As a grader I've given full credit for the wrong answer and no credit for the right answer.
- gizmo686 10y ago
  So, your proposed solution to integral(ln(x)) involves first noticing that d/dx (1/x) = ln(x), then proceeding to use v=1/x to integrate by parts? I think I am misunderstanding something, because as far as I can tell, integration by parts where one of the parts is 1 is literally useless.
  - yequalsx 10y ago
    u = ln(x) and dv = 1 du = 1/x and v = x integral ln(x) = x ln(x) - integral 1/x times x integral ln(x) = x ln(x) - x + C I could give integral arctan(x) but with the advent of computer algebra systems I'm mostly interested in them knowing the basic examples and to not burden them on a test with something more complicated. EDIT: The derivative of 1/x is not ln(x) as you stated. You got it backwards and my guess is that is the source of your confusion.

webmaven 10y ago

> Passing a class should mean more than I got a lot of answers correct. It should mean an understanding of the material. I agree, but testing for understanding (as opposed to Socratically probing for it) is more time consuming, complex, and difficult than just testing for correct answers. To stay within a given test workload, students would have to take far fewer tests. Obviously not the direction the educational system is trending these days. Which is a shame.

Ntrails 10y ago
>I agree, but testing for understanding is more time consuming, complex, and difficult than just testing for correct answers. You get 1 mark for the right answer and 2 marks for working. Problem solved. The question doesn't have to change at all. This is how I remember mark schemes working in the general case
- webmaven 10y ago
  > You get 1 mark for the right answer and 2 marks for working. I'm not sure I get what you mean by "working". Can you expand that a bit?

BeetleB 10y ago

As someone who has spent time in the math world, I sympathize with proactive's response. Once you get to a "high enough" level, the lessons you learned earlier have to be unlearned. Not just true in mathematics but in engineering as well. I was taking an engineering course in my sophomore year where I had a system of equations in 3-4 variables. I spent forever trying to solve it (analytically), and failed. So did most of the class. The next lecture, the professor showed us how to do it. A mixture of plots, etc reduced the solution space and the rest was trivial. He also said "You could just use the solver in your calculator/MATLAB". I wasn't satisfied with his answer. It felt like cheating. I didn't learn the cool way to do things. But in the real world, if you can get the solution this way, it's perfectly valid. As long as you can confirm that you found a/the solution (trivial to do). With the x^3=27 answer, it is the onus of the instructor to specify explicitly that "guessing is not allowed". Why? Because as others have mentioned, it is totally appropriate in mathematical circles to guess a solution. Much work in mathematics is done that way. In various classes (mathematics/engineering/physics), I've both utilized non-standard ways to solve problems on tests, and have seen it done by students on tests I grade. This is to be encouraged. Especially because this is what mathematicians/physicists love to do in their real work. If your goal is to ensure they understand cube roots, either make a problem that is hard to guess (e.g. x^3 = 24), or be explicit about it. Even with x^3 = 24, if they use Newton's method, that should be graded correct.

yequalsx 10y ago
Professional mathematicians don't guess answers to theorems. They sometimes guess counterexamples. But no one guesses answers to a theorem. I've seen, "I notice that A is a solution to this equation does anyone know a method for formally solving it?" Guessing is not a method of solving. It is a method of finding counterexamples. Two different types of problems. I accept any mathematically valid method of solving a problem. Mathematically method means, method that works even if I trivially change it by using different numbers.
- BeetleB 10y ago
  >Professional mathematicians don't guess answers to theorems. Believe it or not, proving theorems is not the goal of many mathematicians. I'm simplifying a bit, but read Freeman Dyson's essay on Birds and Frogs. Essentially "problem solvers" vs "theory builders". While problem solvers often do end up proving theorems, it is not their main goal. If they can "guess" a solution, they are done. It is publishable. Go to the field of combinatorics, and you'll find it is full of guesswork. >I accept any mathematically valid method of solving a problem. Mathematically method means, method that works even if I trivially change it by using different numbers. Sorry, but many mathematicians disagree with you. Solving a problem is finding a solution (provided you have a means to verify correctness). It doesn't matter if you merely guessed it.