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0 pointsproactive9y ago0 comments

You have to be very cautious when marking wrong an answer that is correct, because you're at risk of confusing the student. The guess-and-try approach used in this particular example is very brittle, but it is correct. Professional mathematicians use it frequently, especially with the advent of computer-aided mathematics, for instance to find counter-examples to statements.

I see at least three issues with claiming that answer as wrong: first, correctness is essential (in the true sense) in mathematics, and therefore should not be carelessly dismissed in front of the student. Second, students should not be made to believe that guess-and-try is always inappropriate, but rather to understand that it won't always work. Finally, in this particular example the approach chosen is arguably (at least from the student's perspective) simpler than the one expected by the professor. Invalidating a "simpler" approach might give the student the impression that you always need to take the complicated route (ie, "math is hard") when the opposite is true.

My own take on this example would be to give (partial?) marks, with a lengthy comment of the type "fair enough, in this case, but what about if you wanted to solve x^3 =7? Your method wouldn't work, then!". Alternatively, if you don't want to give marks, it should be justified at length by rules clearly explained before the exam, while acknowledging the correctness of the approach.

0 comments

analog319y ago

Because I can't edit my above post any more, I'll add it here: I marked it correct. I didn't feel like penalizing him for something that had never been taught.

Here's an idea for a better problem:

Solve the following:

x^3 = 27

y^3 = 21

z^4 = 85

My rationale is that there will be a huge time advantage for the student who works out the solutions by using roots, and a visual "hint" that there might be a general solution.

quickben9y ago

At university, they forced us using logs to solve similar equations with extremely large exponents. Otherwise it's nigh impossible to calculate any answer with the allowed calculator.

analog319y ago

I love it. Write numerically unstable problems, so they have to be solved symbolically.

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NumberCruncher9y ago

The biggest advantage for the students would be if you would not split up math into algebra, calculus, etc. The equation

x^3 = 27 (calculating the volume of a cube)

is an application of the equation of

x^y = z (potentiate a number)

which is an application of the equation

f(x) = y (apply a function)

The solution is simply

x = f-inverse(y)

if f is invertible.

As this simple example shows math on high school level can not splitted up into geometry, algebra, calculus, etc. Understanding one of these areas helps to understand the others and vice versa. If you want to master one of them you have to master all of them at the same time, with the same speed, parallel.

yequalsx9y ago

I was under the assumption that this was for a college algebra course and that the rational exponents had been covered. If it hadn't been taught then I would give credit. Indeed, I'd be impressed by such reasoning.

yequalsx9y ago

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.

proactiveOP9y 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.

yequalsx9y 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.

1 more reply

webmaven9y 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.

Ntrails9y 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

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BeetleB9y 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.

yequalsx9y 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.

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analog319y ago

Because I can't edit my above post any more, I'll add it here: I marked it correct. I didn't feel like penalizing him for something that had never been taught.

Here's an idea for a better problem:

Solve the following:

x^3 = 27

y^3 = 21

z^4 = 85

My rationale is that there will be a huge time advantage for the student who works out the solutions by using roots, and a visual "hint" that there might be a general solution.

quickben9y ago

At university, they forced us using logs to solve similar equations with extremely large exponents. Otherwise it's nigh impossible to calculate any answer with the allowed calculator.

analog319y ago

I love it. Write numerically unstable problems, so they have to be solved symbolically.

1 more reply

NumberCruncher9y ago

The biggest advantage for the students would be if you would not split up math into algebra, calculus, etc. The equation

x^3 = 27 (calculating the volume of a cube)

is an application of the equation of

x^y = z (potentiate a number)

which is an application of the equation

f(x) = y (apply a function)

The solution is simply

x = f-inverse(y)

if f is invertible.

yequalsx9y ago

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

proactiveOP9y 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.

yequalsx9y ago

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.

1 more reply

webmaven9y 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.

Ntrails9y 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

1 more reply

BeetleB9y 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.

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).

yequalsx9y ago

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.

2 more replies

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