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.
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.
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.
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.
> 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.
>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
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.
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.
>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.
My training was in commutative algebra as a pure mathematician. I did dabble in combinatorial commutative algebra and went so far as purchase a book on the topic by Sturmfels and Miller. I doubt very much that you can find a published paper in mathematics in which the author guessed the answer to something and did not prove anything. Every paper I read (commutative algebra) proved things or discussed something heuristically with the goal of finding a method of proving.
I have the book by Stanley on combinatorics. There are not results of the form: I guessed A is the answer and it's right. Let's move on. This is does not happen. When one notices something is a solution the mathematician always wonders why. A mathematician wonders what underlying structure there is. Never is one satisfied by a guess.
If someone found a counterexample to the Riemann Hypothesis the first question would be, why is this number a counterexample? What caused the obstruction? You could problem publish a paper that just said, A is a counterexample to the Riemann Hypothesis. But you could not publish a paper that said, I guessed A is a solution to B and it turns out I was right.
Even in commutative algebra, there is a fair amount of guesswork. If I tell you that 2+3=4 in an abelian group with {2,3,4}, then ask you what is 2+4, you will immediately guess 2+4=2. Its just what the millennials term "stupid obvious". Yeah you can do a lengthy proof on why 2+4=2. Proof: Its 2 because 4 must be the additive identity, and 4 is the additive identity because 2 & 3 are not, and they aren't because if 2 was then 2+3=3, and if 3 was then 2+3=2, but because I've told you 2+3=4, by closure 4 must be identity, which implies 2+4=2. That's the whole story. Literally no mathematician I know will sit down & write that lengthy proof I wrote. They'll just tell you 2+4=2, and if you pester them with "But why?" they'll say "Because" and excuse themselves :)
In an elementary group theory course this is the type of problem that would be given when groups are first introduced. It's a good homework/test question and just writing a*c = a as the answer would not suffice (depending on the level of the course and aim of the problem). The point at that level is for the student to learn how to justify their beliefs.
This is especially so if one were in a basic mathematical logic course. Of course, in an algebraic topology course where this group showed up it would be assumed that everyone knows how to find the answer and why. No justification would be needed.
The paper you cited supports what I've been saying. You can publish a paper that says, "Here is a counterexample." You can't publish a paper that says, "Here is a solution I guessed to be correct."
Any method of finding a counterexample is accepted. Guessing a solution is not.
EDIT: The paper linked to was published because it was a counterexample to a famous conjecture.
What is the "answer" to a theorem? If by "answer" you mean the conclusion of the theorem, then mathematicians guess these all the time; such guesses are generally referred to as conjecture.
A conjecture is someone, usually a well known expert in the field, saying, "I think this is true but have not been able to prove it." It's not considered a theorem until someone proves it. A famous example is the Goldbach conjecture. No proof has been found but people have been searching for a proof for a long time but so far no one has proven it.
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.