Assumptions in Mathematical (Calculus) word problems

[Amad27]

You are absolutly right, but this is now a question of communication of the assumptions and not a question of why we need to make assumptions in the first place. Having well-posed mathematical questions is not the same as presenting them in a clear way!

 

 

Actually,

 

Never mind what I said about the Grice Maxim, think about this:

 

As you said the goal is to teach something, so then the problem must be solvable.

 

Ideas?

Edited August 22, 2014 by Amad27

[ajb]

It was because of the grice maxim of quality, that we make the assumptions that what isnt mentioned doesnt exist.

This may tell you why in a textbook when the questions are written down they are of finite length! I mean you don't state assumptions or (typically) include information that is not needed. You don't have to list all the things that do not happen to the balloon!

 

 

Here the answer was because the problems are not "gotcha" problems, they are intended to teach you something. The balloon popping before time, or there being a hole making it impossible to do anything with it.

Exactly, this is why in textbook questions lots of simplifying assumptions and approximations are made.

[Amad27]

This may tell you why in a textbook when the questions are written down they are of finite length! I mean you don't state assumptions or (typically) include information that is not needed. You don't have to list all the things that do not happen to the balloon!

 

 

 

Exactly, this is why in textbook questions lots of simplifying assumptions and approximations are made.

 

Hi,

 

Apparently, I think you posted before I edited my comment.

 

I think you helped me the best with the simplifying assumptions. Can you confirm my thought,

 

They use simplifying assumptions otherwise nothing would be possible.

 

Also, is it a simplifying assumption that there is no gas leak?

 

Thanks ajb, I owe you big, big, big time.

Edited August 22, 2014 by Amad27

[pzkpfw]

If there's a gas leak, but the question hasn't given you the rate of leak, then the problem becomes unsolvable - you'd have to invent the rate of leak, which is pointless. It's not just about simplifying the problem, it's about making it possible to solve. If you want to argue about the semantic difference between "simplification" and "making possible", well, I'm not interested in that.

 

All problems become unsolvable if you get to invent issues not stated. How quickly will an apple fall off a 3 metre high branch? Well, I can't say, because an earthquake might occur the moment the apple drops, which causes the ground to rise at some rate, decreasing the distance the apple has to fall. Silly.

 

Even in the real world, working on real problems, there will be simplifications in any question. You always need to determine which factors don't have enough of an effect to be worth including, and which factors are just unlikely to occur.

[Amad27]

If there's a gas leak, but the question hasn't given you the rate of leak, then the problem becomes unsolvable - you'd have to invent the rate of leak, which is pointless. It's not just about simplifying the problem, it's about making it possible to solve. If you want to argue about the semantic difference between "simplification" and "making possible", well, I'm not interested in that.

 

All problems become unsolvable if you get to invent issues not stated. How quickly will an apple fall off a 3 metre high branch? Well, I can't say, because an earthquake might occur the moment the apple drops, which causes the ground to rise at some rate, decreasing the distance the apple has to fall. Silly.

 

Even in the real world, working on real problems, there will be simplifications in any question. You always need to determine which factors don't have enough of an effect to be worth including, and which factors are just unlikely to occur.

 

Hi,

 

Thanks. This really helped.

 

In the end, in does depend on the context. If the problem is meant to be solved such as in a test or exam correct?

[ajb]

They use simplifying assumptions otherwise nothing would be possible.

This depends on the system, but essentially this is right.

 

Also, is it a simplifying assumption that there is no gas leak?

Yes. As pzkpfw says, without some more information you could not solve the problem with a leak. You would need to know the rate of the gas escaping or something equivalent to that.

[Amad27]

This depends on the system, but essentially this is right.

 

 

Yes. As pzkpfw says, without some more information you could not solve the problem with a leak. You would need to know the rate of the gas escaping or something equivalent to that.

 

Hello @ajb, thanks a bunch =) I appreciate it.

 

I am hearing two things.

 

(1) Assumptions for simplicity

(2) Assumptions for making the problem solvable.

 

I am confused as to, which is correct. =(

[ajb]

I am hearing two things.

 

(1) Assumptions for simplicity

(2) Assumptions for making the problem solvable.

They are not usually distinct.

 

For simplicity, you may be able to argue that some of the variables will not change the situation in any drastic way. Or you may be only interesting in "leading terms" and so can make some approprate approximations.

 

These may or may not help you to solve the problem. It is quite possible that further approximations or simplifying assumptions need to me make for you to actually solve the problem.

[Amad27]

They are not usually distinct.

 

For simplicity, you may be able to argue that some of the variables will not change the situation in any drastic way. Or you may be only interesting in "leading terms" and so can make some approprate approximations.

 

These may or may not help you to solve the problem. It is quite possible that further approximations or simplifying assumptions need to me make for you to actually solve the problem.

 

Thanks @ajb. Very very helpful.

 

When you said,

 

"It is quite possible that further approximations or simplifying assumptions need to me make for you to actually solve the problem."

 

What do you mean by further simplifying assumptions?

 

So really, at the end, you do need to assume the problem is solvable based on the context =) And then make simplifying assumptions based on the problem being solvable? Correct?

[ajb]

What do you mean by further simplifying assumptions?

I don't think it really applies to the sort of questions you are thinking of, but you may model some system with some starting assumptions. You will get a set of equations (whatever kind they are) and although these may be very clear to interpret they may not have exact solutions. From there you could apply some approximation to these equations or see if some further simplifying assmptions at the first stage will help.

 

In your textbooks I would expect you not to have to work quite this hard.

 

So really, at the end, you do need to assume the problem is solvable based on the context.

For the case of textbook questions you can assume there is usually a concrete answer.

 

And then make simplifying assumptions based on the problem being solvable? Correct?

Really, you make assumptions so that you can pose a question that is hopefully solvable.

 

But agian, in the context of textbook questions you typically won't have to worry about this. The assumptions are either given explicitly or will be obvious.

[Amad27]

I don't think it really applies to the sort of questions you are thinking of, but you may model some system with some starting assumptions. You will get a set of equations (whatever kind they are) and although these may be very clear to interpret they may not have exact solutions. From there you could apply some approximation to these equations or see if some further simplifying assmptions at the first stage will help.

 

In your textbooks I would expect you not to have to work quite this hard.

 

 

For the case of textbook questions you can assume there is usually a concrete answer.

 

 

Really, you make assumptions so that you can pose a question that is hopefully solvable.

 

But agian, in the context of textbook questions you typically won't have to worry about this. The assumptions are either given explicitly or will be obvious.

 

Hello again,

 

What about on tests or AP Calculus Exams?

[ajb]

What about on tests or AP Calculus Exams?

I don't know details of AP Exams, but I would expect the assumptions to have been made for you or to be quite obvious. Many of my comments were looking at mathematical modeling at a more advanced level. I don't want you to think that mathematics questions are always easy to pose let alone solve. However, for tests and so on it is the problem of who ever wrote the questions to ensure they are suitable.

[Amad27]

I don't know details of AP Exams, but I would expect the assumptions to have been made for you or to be quite obvious. Many of my comments were looking at mathematical modeling at a more advanced level. I don't want you to think that mathematics questions are always easy to pose let alone solve. However, for tests and so on it is the problem of who ever wrote the questions to ensure they are suitable.

 

Hello =)

 

I want to share something; I think you might be interested.

 

I think in the end it isn't "solvability," but the way we learned to communicate ever since we were children. Suppose me and you have 4 friends.

 

John, Sam, Dolph, Randy

 

I am holding a party at my house and you ask me who I have invited. I simply respond.

 

"I have invited Dolph and Randy."

 

Immediately, what would you assume? Would you think I invited John or Sam?

 

No right? You would think John & Sam are not invited.

 

I suppose it is the SAME here, as long as you TRUST me that I am not stupid.

 

I suppose we have to trust the author, in that he isn't stupid. =)

 

What do you say?

[ajb]

I think in the end it isn't "solvability," but the way we learned to communicate ever since we were children.

I think this is again an issue of the communication of the problem rather than the problem itself.

[Amad27]

I think this is again an issue of the communication of the problem rather than the problem itself.

 

Hello @ajb,

 

So, what do you think the end verdict is here?

 

How should we interpret word problems?

[ajb]

How should we interpret word problems?

As best you can is the best I can really say...

 

Typically enought information will be given, but no more, in order to solve the problem.

[Amad27]

As best you can is the best I can really say...

 

Typically enought information will be given, but no more, in order to solve the problem.

 

Hello @ajb.

 

This is an unclear topic, isn't it?

 

When you say "enough information will be given,"

 

Do you mean this an axiom? Because I see no way to reason this, nor prove this =(

 

Why do you think general high school or college calculus students make such assumptions?

[ajb]

This is an unclear topic, isn't it?

I am sure you are over thinking this.

 

When you say "enough information will be given,"

 

Do you mean this an axiom? Because I see no way to reason this, nor prove this =(

It is not really an axiom. Maybe a metatheorem; a statement about mathematics and not in mathematics.

 

Again, you are really making more of this than there really is.

 

Why do you think general high school or college calculus students make such assumptions?

So that a question can be well-posed and have a solution that the students can reach.

 

I don't see that there is much more to this.

[Amad27]

I am sure you are over thinking this.

 

 

It is not really an axiom. Maybe a metatheorem; a statement about mathematics and not in mathematics.

 

Again, you are really making more of this than there really is.

 

 

So that a question can be well-posed and have a solution that the students can reach.

 

I don't see that there is much more to this.

Hello @ajb.

 

Thanks for everything you have done here.

 

Can I ask you one final question?

 

Lets think away from school. What about just general textbooks. For example Ron Larson, Stewart, Apostol etc..

 

Do you judge the assumptions in the word problems based on context/purpose? Such as, the purpose being to teach mathematics, which can only be done if a problem is solvable?

 

And the context being an informative textbook?

 

What do you say here?

[ajb]

Do you judge the assumptions in the word problems based on context/purpose? Such as, the purpose being to teach mathematics, which can only be done if a problem is solvable?

Generically what I said about high school textbooks will apply to all textbooks. But be aware that a problem not having a well-defined solution can also be very informative and aid teaching. This may fall under "counter examples".

 

I really think you are thinking far to hard about this.

[Amad27]

Generically what I said about high school textbooks will apply to all textbooks. But be aware that a problem not having a well-defined solution can also be very informative and aid teaching. This may fall under "counter examples".

 

I really think you are thinking far to hard about this.

 

Hello @ajb,

 

I know I am thinking too much.

 

But the reason why is because if I don't see a reason to make such assumptions then it is impossible to interpret a word problem and its assumptions.

 

If the problem says "find the rate at, which ...." Then you can safely judge the problem is solvable (numeric solutions etc).

If the problem says "find, if possible, the rate at, which..." then you cannot know if the problem is solvable or not.

 

Really, if a problem is not solvable then you can postulate the existence of holes etc..

On the other hand, if the problem is solvable, then you CANT postulate the existence of holes etc...

 

Aside from that, I know I am over thinking this.

 

How do you suggest I interpret assumptions and word problems WITHOUT over thinking this? Do I just blindly make assumptions (that holes don't exist etc..)?

 

Thanks @ajb. =)

[ajb]

How do you suggest I interpret assumptions and word problems WITHOUT over thinking this? Do I just blindly make assumptions (that holes don't exist etc..)?

Just use the information given and you should not look to further complicate the problem. You could of course comment at the end that you have assumed the balloon does not burst and that it has no holes. I don't think that there would be any reason to state all further assumptions about what does not happen to the balloon as this list would be infinite!

 

For example, one child does not purposely pop the balloon, two children do not purposely pop the balloon, three children do not purposely pop the balloon...

 

You just need to keep it simple as possible.

[Amad27]

Just use the information given and you should not look to further complicate the problem. You could of course comment at the end that you have assumed the balloon does not burst and that it has no holes. I don't think that there would be any reason to state all further assumptions about what does not happen to the balloon as this list would be infinite!

 

For example, one child does not purposely pop the balloon, two children do not purposely pop the balloon, three children do not purposely pop the balloon...

 

You just need to keep it simple as possible.

 

True. It would be impossible for the author to

(A) Write every assumptions

(B) For us to write every assumptions.

 

The only doubt is that, we can never know if the author wants the uncomplicated or the complicated.

 

What a shame.

 

Thanks @ajb, your comments really really really help.

[ajb]

The only doubt is that, we can never know if the author wants the uncomplicated or the complicated.

The questions are usually written so that you use the information in the question only, apart maybe from mathematical and physical constants that you can simply look up. So again, don't make life harder than it needs to be when approaching word questions.

[Amad27]

The questions are usually written so that you use the information in the question only, apart maybe from mathematical and physical constants that you can simply look up. So again, don't make life harder than it needs to be when approaching word questions.

 

Hello @ajb,

 

When you said,

 

"The questions are usually written so that you use the information in the question only"

 

The whole point of my issue was how do you know that?

 

Again, is it axiomatic or a theory?