Assumptions in Math (Calculus) word problems

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Amad27
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Assumptions in Math (Calculus) word problems

Postby Amad27 » Tue Aug 19, 2014 11:34 am UTC

Hello there,

I just had a small question about assumptions in mathematical word problems. Suppose you are given a calculus problem (related-rates),

"A spherical balloon is inflated with gas at the
rate of 800 cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is (a) 30 centimeters and (b) 60 centimeters?"
(Larson Calculus P 153)

This is a quoted problem from Larson's Calculus 10th Edition.

My question is here:

Why do you assume for example that there is no hole from, which air LEAVES?
Basically, in general, why do you make unstated assumptions for example, there is no air leaving, or the balloon doesnt explode before the radius is (a) 30 cm etc..?

Here is what others say,

Others say it is so you could solve the problem, what do you think?

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Re: Assumptions in Math (Calculus) word problems

Postby arbiteroftruth » Wed Aug 20, 2014 3:32 am UTC

The phrasing "the balloon is inflated at 800 cm3/min" directly refers to the rate at which the amount of air inside the balloon is increasing. It doesn't matter whether that value is achieved with 800 in and 0 out, or 1200 in and 400 out, or whatever. In any such case, the *balloon* is inflating at 800 cm3/min, even if the total amount of air flow involved is different.

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Re: Assumptions in Math (Calculus) word problems

Postby Forest Goose » Wed Aug 20, 2014 3:55 am UTC

If you were interested in actual balloons and filling them with air, then what you say would be something to consider.

The topic in the book - what it is you are interested in - is not balloons, but related rates. The intent of the problem is not to investigate balloon phenomena, but to teach the concept. The reason you are supposed to assume that everything works out nicely is because there is an obvious problem they want you to solve. Assuming the balloon will burst is not really germane to learning related rates - the textbook is not on the manufacture of balloons, but calculus.

In pedagogical cases, worrying over off topic details is like criticizing analogies on grounds not related - if I said "Doctors investigate and diagnosis disease like a detective on a case", it would be odd to say, "Not really, detectives get paid less and don't need malpractice insurance." It's not that that's not true, nor that that might never be relevant to something, but it is missing the entire point of what was being illustrated by the analogy. Your word problem is similar, it is trying to communicate, "Solve this problem about related relates", not "Determine how balloons function."
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Re: Assumptions in Math (Calculus) word problems

Postby ConMan » Wed Aug 20, 2014 4:02 am UTC

Amad27 wrote:My question is here:

Why do you assume for example that there is no hole from, which air LEAVES?
Basically, in general, why do you make unstated assumptions for example, there is no air leaving, or the balloon doesnt explode before the radius is (a) 30 cm etc..?


Because the biggest assumption that you need to make is that the problem already includes all of the relevant data you need to solve the problem. Otherwise, that's when you get into the area of saying "The ball on the slope doesn't roll because Superman is holding it in place."

The other reason is that the question is not trying to get you to find the smart-alec, technically correct answer. It's trying to get you to apply a method, or a formula, and it is providing a situation in which that method applies.

Sometimes, you do not have all of the data, and this is either because there was an error in setting up the question or there is an expectation that you already know something else, or might be expected to notice that the data is missing and will look it up. But even then, those are questions where one of the skills it's trying to get you to apply is the critical thinking to notice that you need to know something else, and work out how to find it - and it's likely that somewhere in or near the question is something that will help point you in the right direction.
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Re: Assumptions in Math (Calculus) word problems

Postby Meteoric » Wed Aug 20, 2014 7:56 am UTC

Normal human communication usually obeys the Gricean maxims. Specifically applicable here is the maxim of quantity.

For example: I'm late to work, and my boss calls me to ask where I am. "My radiator burst and I had to call a tow truck,"I tell him. In normal conversation, my statement would be taken as the reason I am not at work. My statement is also completely consistent with having just overslept and, as a non sequitur, bringing up the time my radiator burst two years ago - but nobody will think I mean that, and in fact most people would call me a liar if I did, even though every word I said was technically true. This is why you can lie by omission - by the normal rules of conversation, saying anything implies "and this is relevant and I have not omitted anything else relevant". Allowing those implications to stand is dishonest, even though you didn't say them out loud.

Math problems work the same way. If the problem tells you about inflation and asks you about a radius, but does not tell you about a hole in the balloon or ask you about popping, you can assume that the balloon does not have extra holes and does not pop. You should instead assume that inflation occurs as described until at least 60cm, because if it doesn't, the problem has lied to you.
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Re: Assumptions in Math (Calculus) word problems

Postby Amad27 » Wed Aug 20, 2014 8:46 am UTC

Helloooo,


All of your replies were VERY helpful. THANK YOU!

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Re: Assumptions in Math (Calculus) word problems

Postby Amad27 » Wed Aug 20, 2014 12:09 pm UTC

Meteoric wrote:Normal human communication usually obeys the Gricean maxims. Specifically applicable here is the maxim of quantity.

For example: I'm late to work, and my boss calls me to ask where I am. "My radiator burst and I had to call a tow truck,"I tell him. In normal conversation, my statement would be taken as the reason I am not at work. My statement is also completely consistent with having just overslept and, as a non sequitur, bringing up the time my radiator burst two years ago - but nobody will think I mean that, and in fact most people would call me a liar if I did, even though every word I said was technically true. This is why you can lie by omission - by the normal rules of conversation, saying anything implies "and this is relevant and I have not omitted anything else relevant". Allowing those implications to stand is dishonest, even though you didn't say them out loud.

Math problems work the same way. If the problem tells you about inflation and asks you about a radius, but does not tell you about a hole in the balloon or ask you about popping, you can assume that the balloon does not have extra holes and does not pop. You should instead assume that inflation occurs as described until at least 60cm, because if it doesn't, the problem has lied to you.


This is particularly VERY interesting. The concept of Cooperative principle could be a very good reason.

This follows,

"The maxim of quality, where one tries to be truthful, and does not give information that is false or that is not supported by evidence."
Very good, you are awesome. This could be my problem solved.

I have a question for everyone.

Why do you think they never teach these maxim of quality idea in school when you LEARN to solve word problems?
Thanks

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Re: Assumptions in Math (Calculus) word problems

Postby ahammel » Wed Aug 20, 2014 1:52 pm UTC

ConMan wrote:Because the biggest assumption that you need to make is that the problem already includes all of the relevant data you need to solve the problem. Otherwise, that's when you get into the area of saying "The ball on the slope doesn't roll because Superman is holding it in place."
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Re: Assumptions in Math (Calculus) word problems

Postby LaserGuy » Wed Aug 20, 2014 2:44 pm UTC

Amad27 wrote:Why do you think they never teach these maxim of quality idea in school when you LEARN to solve word problems?
Thanks


I'd be tempted to say that it's a well-understood rule of normal conversation that people operate under all the time. Even if they couldn't specifically name the principle, it is generally understood in practice because it's essential to functional communication.

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Re: Assumptions in Math (Calculus) word problems

Postby Amad27 » Wed Aug 20, 2014 3:07 pm UTC

LaserGuy wrote:
Amad27 wrote:Why do you think they never teach these maxim of quality idea in school when you LEARN to solve word problems?
Thanks


I'd be tempted to say that it's a well-understood rule of normal conversation that people operate under all the time. Even if they couldn't specifically name the principle, it is generally understood in practice because it's essential to functional communication.


Thank you.

So the cooperative principle is really the rule of communication?

Thought, it can be broken.

Really, in another forum, someone else said, without making these assumptions it is impossible to do anything.

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Re: Assumptions in Math (Calculus) word problems

Postby brenok » Wed Aug 20, 2014 7:54 pm UTC

Amad27 wrote:Thank you.

So the cooperative principle is really the rule of communication?

Thought, it can be broken.

Really, in another forum, someone else said, without making these assumptions it is impossible to do anything.

Of course, if you can't expect good faith between the examiner and the examinee questions would have to be infinitely long.

"Solve the following problem, assuming there is no gravity, and friction is neglegible, and there is no air, and neglecting relativistic and quantum effects, and assuming there isn't an elephant or any large mammalian in the way, or a superhero isn't blocking any step of the mechanism, and that we are using the decimal system, and the language used is English, and assume all information on this question is true, and that the previous sentence is not false..."

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Re: Assumptions in Math (Calculus) word problems

Postby Bloopy » Thu Aug 21, 2014 4:08 am UTC

ConMan wrote:Because the biggest assumption that you need to make is that the problem already includes all of the relevant data you need to solve the problem.

This. The Wikipedia article on problem solving may also be useful.

Ignoring the possibility that the balloon pops isn't much of an assumption. You know for a fact that if you insist it pops at 25cm, then you'll have no answers to the questions. So you can take it as hypothetical and do the calculation anyway (regardless of whether every balloon in existence is known to pop and/or get stomped on by an elephant upon reaching 25cm).

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Re: Assumptions in Math (Calculus) word problems

Postby Amad27 » Thu Aug 21, 2014 8:16 am UTC

Bloopy wrote:
ConMan wrote:Because the biggest assumption that you need to make is that the problem already includes all of the relevant data you need to solve the problem.

This. The Wikipedia article on problem solving may also be useful.

Ignoring the possibility that the balloon pops isn't much of an assumption. You know for a fact that if you insist it pops at 25cm, then you'll have no answers to the questions. So you can take it as hypothetical and do the calculation anyway (regardless of whether every balloon in existence is known to pop and/or get stomped on by an elephant upon reaching 25cm).


So really,

You must make certain assumptions to be able to SOLVE the problem?

Why do they never teach this in textbooks?

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Re: Assumptions in Math (Calculus) word problems

Postby LaserGuy » Thu Aug 21, 2014 3:06 pm UTC

In any given problem, if you had to spell out all of the assumptions being made, the problem itself would be needlessly long and complex to read. Rather than one or two lines, each problem might be a page or two, or, depending on how deep into epistemology you want to go, perhaps many pages in length. This does little to help the objective of the text, which is to teach you calculus. If anything, it obfuscates what is actually important with many things that aren't. That doesn't mean that it isn't a good exercise to think about what assumptions and approximations you are making in a problem and how valid they are and when they might break down, and indeed, you do find some texts that discuss this in some detail when it is often pedagogically relevant to do so--if you're reading a physics text, for example, they may well preface certain sections with notation that they talking about situations that occur well below the speed of light, because as it will turn out, when the text goes on to discuss relativity, several assumptions related to how we think about time and space break down at high velocities. Thus it is pedagogically relevant to herald the breakdown of those assumptions earlier in the text. But they won't bother to discuss the assumption, say, that induction works, or that the universe is parsimonious, or even logically consistent, despite the fact that these are significant and philosophically challenging assumptions to make. On the other hand, a philosophy of science text will probably go into these assumptions and the problems with them in extensive detail.

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Re: Assumptions in Math (Calculus) word problems

Postby gmalivuk » Thu Aug 21, 2014 3:08 pm UTC

Amad27 wrote:Why do they never teach this in textbooks?

Because by the time you enter school, you have long since internalized the lesson that you have to make certain assumptions about the world in order to interact with it at all, including certain assumptions about things other people write or say in order to communicate effectively.
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Re: Assumptions in Math (Calculus) word problems

Postby firechicago » Thu Aug 21, 2014 3:27 pm UTC

Amad27 wrote:So really,

You must make certain assumptions to be able to SOLVE the problem?

Why do they never teach this in textbooks?

To elaborate on gmal's answer with an example:

You have to make these same sort of assumptions to interpret any communication. For example, if I'm looking for someone I haven't met before, and I ask you "How will I recognize her?" and you answer "She was wearing a red shirt." I have to interpret that statement as being relevant to the question that I have asked. In order to be able to make any sense of it at all. If you were to say "She was wearing a red shirt" but leave out the fact that this was 6 months ago, or that she was wearing a red shirt but said that she was going to change into a different shirt, I would quite reasonably consider you to have lied to me, because, while the statement taken out of context was true, the way in which it was true was not responsive to the context in which it was said. It would be a lie because it appeared to give me the information I was asking for, but in fact did nothing of the sort.

Similarly, we presume that question writers in textbooks are not lying in this way, that they are giving us all the necessary and sufficient information. To explicitly state this assumption is overkill, since it is the same assumption that we make every time that we interpret any statement that anyone makes.

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Re: Assumptions in Math (Calculus) word problems

Postby Amad27 » Thu Aug 21, 2014 4:32 pm UTC

firechicago wrote:
Amad27 wrote:So really,

You must make certain assumptions to be able to SOLVE the problem?

Why do they never teach this in textbooks?

To elaborate on gmal's answer with an example:

You have to make these same sort of assumptions to interpret any communication. For example, if I'm looking for someone I haven't met before, and I ask you "How will I recognize her?" and you answer "She was wearing a red shirt." I have to interpret that statement as being relevant to the question that I have asked. In order to be able to make any sense of it at all. If you were to say "She was wearing a red shirt" but leave out the fact that this was 6 months ago, or that she was wearing a red shirt but said that she was going to change into a different shirt, I would quite reasonably consider you to have lied to me, because, while the statement taken out of context was true, the way in which it was true was not responsive to the context in which it was said. It would be a lie because it appeared to give me the information I was asking for, but in fact did nothing of the sort.

Similarly, we presume that question writers in textbooks are not lying in this way, that they are giving us all the necessary and sufficient information. To explicitly state this assumption is overkill, since it is the same assumption that we make every time that we interpret any statement that anyone makes.


Hello.

This and gmal really helped. (ALOT).

So in the end, really it is a true fact (AXIOM) that the writers of the problem wouldn't lie about something.

Hey also, can you look at @Meteoric's post about Grice Maxim's? Perhaps you could comment on that.
Thanks

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Re: Assumptions in Math (Calculus) word problems

Postby gmalivuk » Thu Aug 21, 2014 6:06 pm UTC

What's to comment about? Grice did a pretty good job pinning down the typical assumptions we all make in cooperative conversations.

Notice the cooperative part, though. One reason legal documents are often so opaque to most people is that legal writing is based on the assumption of adversarial communication. Lots more details and contingencies have to be written into contracts because some people will try to find loopholes and get out of it.

If I expected math students to routinely add their own irrelevant complicating factors to a problem, I'd probably specify a lot more, and then a one-sentence story problem would become a full legal agreement.

(Of course, in some cases it's important to gauge a student's ability to pick out what is important, in which case irrelevant details will be included, or ability to anticipate possible complicating factors, in which case the "correct" answer might depend on whether the student thought of any of those possibilities.)
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Re: Assumptions in Math (Calculus) word problems

Postby Amad27 » Thu Aug 21, 2014 6:12 pm UTC

gmalivuk wrote:What's to comment about? Grice did a pretty good job pinning down the typical assumptions we all make in cooperative conversations.



Hello,

I took this part because it fits the issue. Thanks a lot =)

Would it be considered a cooperative cooperation in for example textbook problems or even AP Exam questions (for calculus eg.) ??

Probably, because the author and us are working to the same end, which is finding a solution.

The only flaw is that this is not an absolute rule, it CAN be broken.

But as the wikipedia page of "cooperative principle" states,

We often assume the speaker is following the Grice Maxims.

Is it safe to always assume that in word problems?

Thanks gmal.

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Re: Assumptions in Math (Calculus) word problems

Postby BedderDanu » Fri Aug 22, 2014 7:42 am UTC

Not necessarily.

In my beginning engineering classes, we got problems like your example. Then, as the classes got harder, they started asking things like "List the assumptions you had to make when solving this problem." Then, in a strength of materials class, A question like this might be part of a multi-problem thing where it would go "Calculate the rate of change of surface tension of the balloon with respect to radius. Then, assuming that the balloon is x mills thick, calculate the maximum radius of the balloon before bursting. (see fig 37 for material rupture stress)."

At the end of it, they have you questioning a lot of the assumptions of a problem, or forcing you to look up values outside of the question. This prepares you for the shift: "Design a balloon that can inflate to 60cm using an 800 mL/min source of STP air."

To throw back to the original question, why did you assume that the proper response to the question was solving the math problem? Why not correct the Grammar? Sure, it was in a calculus book, but why did that matter?

At the end of the day, the goal is to learn something, in this case related rates. The assumptions you make are supposed to help you with that goal. Assuming the balloon bursts does not help you learn related rates. Assuming there is a leak turns the problem into a differential equation (most leaks are based on pressure, which would be based loosely on the radius of the balloon), so not related rates.

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Re: Assumptions in Math (Calculus) word problems

Postby Meteoric » Fri Aug 22, 2014 10:48 am UTC

Amad27 wrote:We often assume the speaker is following the Grice Maxims.

Is it safe to always assume that in word problems?

Yes, but with a couple caveats.

Word problems are not a special case, here. For all communication, Grice's maxims are usually obeyed. There are also some specific ways to break those rules, such that cooperation does not break down. An obvious example is sarcasm, which breaks the maxims of quality and/or manner, but with contextual cues (tone of voice, etc) so that the other person still knows what you really mean. Saying untrue things without sarcasm cues is called "lying".

There are also math problems that break Grice's maxims. We call these "trick questions", and they are a very important teaching tool, so you absolutely should not assume that you will never face a trick question. But like sarcasm, there are some rules for breaking the rules. A few common categories of trick questions are:
-Extraneous information - The problem violates the maxim of quantity, providing more information than is required. This teaches you the skill of recognizing which information is actually relevant.
-Incomplete information - The question violates the maxim of quantity in the other direction, not providing enough information to solve the problem. This teaches the skill of knowing which information you need.
-Contradictory information - The question as posed involves an impossibility, violating the maxim of quality. This teaches the skill of noticing contradiction.

In the latter two cases, you can't directly answer the question as asked! But this really just means that the correct answer is "not enough information" or "that's impossible", not that the question is unanswerable. Even though they violate the maxims, these questions are still cooperative, because they are solvable. Also, questions like this will sometimes give warning, like instructions that say "solve, if possible".

A non-cooperative trick question is just a gotcha, as in "Gotcha! The balloon pops at 25cm!" You can generally assume that math questions are not gotchas, because gotchas don't teach you anything about math.

One final note: sometimes, the person writing the question makes a mistake, making the problem contradictory or incomplete even though it isn't supposed to be! These can be especially difficult to spot, because they won't have any this-might-be-a-wrong-question cues.
Amad27 wrote:Why do they never teach this in textbooks?

For the same reason your math book doesn't teach you to read from left to right. It's outside the scope of the textbook, it's not specific to math, and it's something you should already know. A book specifically on problem-solving might mention it, though.

It's one of those things so fundamental most people don't know they know it. Personally, I think it's a little weird that we teach students formal logical implication but don't even tell them that normal implication works differently; knowing about Grice's maxims can be useful.
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Re: Assumptions in Math (Calculus) word problems

Postby Amad27 » Fri Aug 22, 2014 12:40 pm UTC

Meteoric wrote:
Amad27 wrote:We often assume the speaker is following the Grice Maxims.

Is it safe to always assume that in word problems?

Yes, but with a couple caveats.

Word problems are not a special case, here. For all communication, Grice's maxims are usually obeyed. There are also some specific ways to break those rules, such that cooperation does not break down. An obvious example is sarcasm, which breaks the maxims of quality and/or manner, but with contextual cues (tone of voice, etc) so that the other person still knows what you really mean. Saying untrue things without sarcasm cues is called "lying".

There are also math problems that break Grice's maxims. We call these "trick questions", and they are a very important teaching tool, so you absolutely should not assume that you will never face a trick question. But like sarcasm, there are some rules for breaking the rules. A few common categories of trick questions are:
-Extraneous information - The problem violates the maxim of quantity, providing more information than is required. This teaches you the skill of recognizing which information is actually relevant.
-Incomplete information - The question violates the maxim of quantity in the other direction, not providing enough information to solve the problem. This teaches the skill of knowing which information you need.
-Contradictory information - The question as posed involves an impossibility, violating the maxim of quality. This teaches the skill of noticing contradiction.

In the latter two cases, you can't directly answer the question as asked! But this really just means that the correct answer is "not enough information" or "that's impossible", not that the question is unanswerable. Even though they violate the maxims, these questions are still cooperative, because they are solvable. Also, questions like this will sometimes give warning, like instructions that say "solve, if possible".

A non-cooperative trick question is just a gotcha, as in "Gotcha! The balloon pops at 25cm!" You can generally assume that math questions are not gotchas, because gotchas don't teach you anything about math.

One final note: sometimes, the person writing the question makes a mistake, making the problem contradictory or incomplete even though it isn't supposed to be! These can be especially difficult to spot, because they won't have any this-might-be-a-wrong-question cues.
Amad27 wrote:Why do they never teach this in textbooks?

For the same reason your math book doesn't teach you to read from left to right. It's outside the scope of the textbook, it's not specific to math, and it's something you should already know. A book specifically on problem-solving might mention it, though.

It's one of those things so fundamental most people don't know they know it. Personally, I think it's a little weird that we teach students formal logical implication but don't even tell them that normal implication works differently; knowing about Grice's maxims can be useful.


I think the problem is solved.

So really, it was all about communication. Have you realized something?

You learn Grice Maxims implicitly
You learn it but dont know what it specifically is.

A last question, about the "gotchas"

On tests, what is a supposed reason to assume "non-gotchas" ?

Thanks =)

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Re: Assumptions in Math (Calculus) word problems

Postby brenok » Fri Aug 22, 2014 12:48 pm UTC

Because gotchas don't teach you anything about Math?

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Re: Assumptions in Math (Calculus) word problems

Postby Amad27 » Fri Aug 22, 2014 1:16 pm UTC

brenok wrote:Because gotchas don't teach you anything about Math?


Hello,

Thanks for this.

But tests arent supposed to teach, supposed to test skill

Thoughts?

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Re: Assumptions in Math (Calculus) word problems

Postby LaserGuy » Fri Aug 22, 2014 2:46 pm UTC

Amad27 wrote:
brenok wrote:Because gotchas don't teach you anything about Math?


Hello,

Thanks for this.

But tests arent supposed to teach, supposed to test skill

Thoughts?


There's no reason that you can't design a test both to teach and to test ability. This is commonly referred to in education theory as "assessment as learning" (as opposed to "assessment of learning", which is the more traditional summative exam at the end of a unit or course).

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Re: Assumptions in Math (Calculus) word problems

Postby brenok » Fri Aug 22, 2014 3:51 pm UTC

Amad27 wrote:
brenok wrote:Because gotchas don't teach you anything about Math?


Hello,

Thanks for this.

But tests arent supposed to teach, supposed to test skill

Thoughts?

Trying to cheat the system by pointless semantics and details is usually not part of the skillset tested on math exams.

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Re: Assumptions in Math (Calculus) word problems

Postby gmalivuk » Fri Aug 22, 2014 9:24 pm UTC

Amad27 wrote:
brenok wrote:Because gotchas don't teach you anything about Math?


Hello,

Thanks for this.

But tests arent supposed to teach, supposed to test skill

Thoughts?

Gotchas also don't test whether you understand the math being assessed.

I can't check whether or not you understand the mathematics of related rates if I try to trick you by assuming that the balloon pops before it gets big enough.
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Re: Assumptions in Math (Calculus) word problems

Postby Tirian » Sat Aug 23, 2014 6:12 am UTC

At the same time, there is a place in mathematical assessment to check if students understand the limitations of models that is as important as checking to see if they can apply the model. If I told you that I inflated a child's balloon 10 cubic centimeters in the first second of inflating it and asked what the volume of the balloon would be after ten minutes of inflation at this rate, you would not be demonstrating a comprehension of proportional reasoning by plugging numbers into the formula without thought.

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Re: Assumptions in Math (Calculus) word problems

Postby Forest Goose » Sat Aug 23, 2014 8:49 am UTC

Tirian wrote:At the same time, there is a place in mathematical assessment to check if students understand the limitations of models that is as important as checking to see if they can apply the model. If I told you that I inflated a child's balloon 10 cubic centimeters in the first second of inflating it and asked what the volume of the balloon would be after ten minutes of inflation at this rate, you would not be demonstrating a comprehension of proportional reasoning by plugging numbers into the formula without thought.


For what type of class would you put this on a test, though? I can see, perhaps, mentioning it, but in a mathematics course on Calculus, wouldn't the natural assumption to make be that the model wasn't really thought out and exists for the sake of the problem? To be perfectly honest, I have no good concept of how inflated such a balloon would be from what you said, nor of how that relates to the capacity of a child's balloon - I'm not sure why a mathematics course should assume that I do.
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Re: Assumptions in Math (Calculus) word problems

Postby Amad27 » Sat Aug 23, 2014 1:49 pm UTC

Forest Goose wrote:
Tirian wrote:At the same time, there is a place in mathematical assessment to check if students understand the limitations of models that is as important as checking to see if they can apply the model. If I told you that I inflated a child's balloon 10 cubic centimeters in the first second of inflating it and asked what the volume of the balloon would be after ten minutes of inflation at this rate, you would not be demonstrating a comprehension of proportional reasoning by plugging numbers into the formula without thought.


For what type of class would you put this on a test, though? I can see, perhaps, mentioning it, but in a mathematics course on Calculus, wouldn't the natural assumption to make be that the model wasn't really thought out and exists for the sake of the problem? To be perfectly honest, I have no good concept of how inflated such a balloon would be from what you said, nor of how that relates to the capacity of a child's balloon - I'm not sure why a mathematics course should assume that I do.


Actually, it is not about being solvable or not.

It is how we have learned to communicate ever since we were young. Suppose you and I have four common friends.

Jon, Jim, Brock, Sam.

I am holding a party at my house, and I tell you, I have invited Jon and Brock. What would you assume? I am confident that you will immediately think that I have NOT invited Jim or Sam.

The critical question is why? For the people you know are serious, you communicate and interpret the way I have written above. Now,

If I said that

I have invited Jon and Brock, and I LAUGH immediately after that, you will know that I am HIDING something.

It is a similar idea, which works in every subject. Physics, Chemistry, Biology, Mathematics, even Reading. All of those have word problems, which use a very similar technique of interpretation.

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Re: Assumptions in Math (Calculus) word problems

Postby Tirian » Sat Aug 23, 2014 2:11 pm UTC

One of the major "fronts" of the math wars is just this: the balance between procedural fluency and conceptual understanding. Traditionally, the prevailing attitude has been the one you stated, paying lip service to conceptual understanding but basing our assessments only on how well students can follow the recipes. The consequences of this decision are as easy to predict as they are to observe in society: a workforce that is under-prepared for rapid shifts in technology and on-site problem solving. One also hears frequent anecdotal evidence that adults feel that they don't use their math skills in the real world, which is hardly surprising when our lessons and tests stick to highly artificial and idealized parameters.

Calculus wasn't invented to fill a slot in the core of a liberal arts education. It was invented to provide a useful abstract model for phenomena in physics and engineering (that turns out to have applications in finance, statistics, and other fields throughout the scientific spectrum). I agree that a careful analysis of the errors introduced from simplistic modeling is difficult. But what's the use of teaching calculus to non-majors if it will only train them to solve problems in an impossibly idealistic universe?

To anyone who has twelve minutes to spare, I highly recommend this TED talk by Dan Meyer, in which he discusses alternatives to teaching problem-solving where students are given all the relevant information and the formula:
https://www.youtube.com/watch?v=BlvKWEvKSi8

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Re: Assumptions in Math (Calculus) word problems

Postby brenok » Sat Aug 23, 2014 2:38 pm UTC

You really sound like someone who when asked "could you pass me the salt" would answer "yes I could" and keep staring at them until they grudgingly phrasethings more explicitly.

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Re: Assumptions in Math (Calculus) word problems

Postby Forest Goose » Sat Aug 23, 2014 3:11 pm UTC

Tirian wrote:One of the major "fronts" of the math wars is just this: the balance between procedural fluency and conceptual understanding. Traditionally, the prevailing attitude has been the one you stated, paying lip service to conceptual understanding but basing our assessments only on how well students can follow the recipes. The consequences of this decision are as easy to predict as they are to observe in society: a workforce that is under-prepared for rapid shifts in technology and on-site problem solving. One also hears frequent anecdotal evidence that adults feel that they don't use their math skills in the real world, which is hardly surprising when our lessons and tests stick to highly artificial and idealized parameters.

Calculus wasn't invented to fill a slot in the core of a liberal arts education. It was invented to provide a useful abstract model for phenomena in physics and engineering (that turns out to have applications in finance, statistics, and other fields throughout the scientific spectrum). I agree that a careful analysis of the errors introduced from simplistic modeling is difficult. But what's the use of teaching calculus to non-majors if it will only train them to solve problems in an impossibly idealistic universe?

To anyone who has twelve minutes to spare, I highly recommend this TED talk by Dan Meyer, in which he discusses alternatives to teaching problem-solving where students are given all the relevant information and the formula:
https://www.youtube.com/watch?v=BlvKWEvKSi8


I'm not saying that we should, or should not, ask such questions, but that if such a question is asked on a math exam, it shouldn't be assumed that I'll respond "the balloon will burst". I don't know anything about balloons, why should I?

More to the point, I don't think people not being forced to second guess math tests is why people can't adapt, I think that the fact that technology is more omnipresent than ever and advancing faster than ever might have something to do with that - it's a hell of a lot harder to keep up with tech solutions than it was fifty years ago, at some point, the average person is going to have trouble adapting.

But, might it not be better to, I don't know, have classes that teach applications and problem solving instead of trying to use tricky details about whether, or not, the balloon in question is a "children's" balloon?

And, at which point do we not put up with people treating problems as if they were "real world"? If I make a test that says give me an example of a turing machine that recognizes the primes, would it be a legitimate answer for a student to write, "Use the C++ library X instead.". That sounds stupid, at first, but it is far more practical, far more reasonable, and far more like what you would actually do; yet, I'm pretty sure no one would accept that as a legit answer to the problem. So why should people worry over how much a balloon can inflate?

Finally, a mathematics course should teach mathematics, I'm not sure why it should be teaching critical thinking. If a student hasn't learned any critical thinking by that point, tricky calc. questions aren't going to teach it to them; and if they do think critically, I imagine they'll wonder why they're expected to consider a balloon's tolerance to inflation since it has no practical point for them either. Who benefits from these kinds of questions? Contrary students who want to pat themselves on the back for being clever enough to spot trick questions and point them out, rather than solve the problem. Seriously, I've tutored a large number of mathematics students, a lot of the one's who don't get the concepts (but don't want to admit it) are usually the same one's that point out objections to the problems along these lines; they are not the "best prepared", they are the worst prepared and take the longest to get to a point where they understand (and I know they don't understand because they can't do anything with problems that aren't word problems since there is nothing to object to). [Last part is personal experience, discount it if you want.]

@Amad: Honestly, I don't have a anything beyond the barest clue what your response means. Would you mind clarifying? I especially don't get what laughing after talking about your party buddy list has to do with physics questions posed in a pedagogical context.
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Re: Assumptions in Math (Calculus) word problems

Postby Amad27 » Sat Aug 23, 2014 3:34 pm UTC

Forest Goose wrote:
Tirian wrote:One of the major "fronts" of the math wars is just this: the balance between procedural fluency and conceptual understanding. Traditionally, the prevailing attitude has been the one you stated, paying lip service to conceptual understanding but basing our assessments only on how well students can follow the recipes. The consequences of this decision are as easy to predict as they are to observe in society: a workforce that is under-prepared for rapid shifts in technology and on-site problem solving. One also hears frequent anecdotal evidence that adults feel that they don't use their math skills in the real world, which is hardly surprising when our lessons and tests stick to highly artificial and idealized parameters.

Calculus wasn't invented to fill a slot in the core of a liberal arts education. It was invented to provide a useful abstract model for phenomena in physics and engineering (that turns out to have applications in finance, statistics, and other fields throughout the scientific spectrum). I agree that a careful analysis of the errors introduced from simplistic modeling is difficult. But what's the use of teaching calculus to non-majors if it will only train them to solve problems in an impossibly idealistic universe?

To anyone who has twelve minutes to spare, I highly recommend this TED talk by Dan Meyer, in which he discusses alternatives to teaching problem-solving where students are given all the relevant information and the formula:
https://www.youtube.com/watch?v=BlvKWEvKSi8


I'm not saying that we should, or should not, ask such questions, but that if such a question is asked on a math exam, it shouldn't be assumed that I'll respond "the balloon will burst". I don't know anything about balloons, why should I?

More to the point, I don't think people not being forced to second guess math tests is why people can't adapt, I think that the fact that technology is more omnipresent than ever and advancing faster than ever might have something to do with that - it's a hell of a lot harder to keep up with tech solutions than it was fifty years ago, at some point, the average person is going to have trouble adapting.

But, might it not be better to, I don't know, have classes that teach applications and problem solving instead of trying to use tricky details about whether, or not, the balloon in question is a "children's" balloon?

And, at which point do we not put up with people treating problems as if they were "real world"? If I make a test that says give me an example of a turing machine that recognizes the primes, would it be a legitimate answer for a student to write, "Use the C++ library X instead.". That sounds stupid, at first, but it is far more practical, far more reasonable, and far more like what you would actually do; yet, I'm pretty sure no one would accept that as a legit answer to the problem. So why should people worry over how much a balloon can inflate?

Finally, a mathematics course should teach mathematics, I'm not sure why it should be teaching critical thinking. If a student hasn't learned any critical thinking by that point, tricky calc. questions aren't going to teach it to them; and if they do think critically, I imagine they'll wonder why they're expected to consider a balloon's tolerance to inflation since it has no practical point for them either. Who benefits from these kinds of questions? Contrary students who want to pat themselves on the back for being clever enough to spot trick questions and point them out, rather than solve the problem. Seriously, I've tutored a large number of mathematics students, a lot of the one's who don't get the concepts (but don't want to admit it) are usually the same one's that point out objections to the problems along these lines; they are not the "best prepared", they are the worst prepared and take the longest to get to a point where they understand (and I know they don't understand because they can't do anything with problems that aren't word problems since there is nothing to object to). [Last part is personal experience, discount it if you want.]

@Amad: Honestly, I don't have a anything beyond the barest clue what your response means. Would you mind clarifying? I especially don't get what laughing after talking about your party buddy list has to do with physics questions posed in a pedagogical context.


Hello,

You didn't understand the point of my last post.

The point is math requires communication.

The point of "requires communication, is that you must use communication skills, while interpreting the word problems.

The "interpreting the word problems" means when something isnt stated, it doesnt exist.

Like the example of the who I invited to my "party."

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Re: Assumptions in Math (Calculus) word problems

Postby Forest Goose » Sat Aug 23, 2014 3:48 pm UTC

Amad27 wrote:Hello,

You didn't understand the point of my last post.

The point is math requires communication.

The point of "requires communication, is that you must use communication skills, while interpreting the word problems.

The "interpreting the word problems" means when something isnt stated, it doesnt exist.

Like the example of the who I invited to my "party."


Why are you bolding things at me, the random bolded phrases don't really convey any extra information, they just look kooky.

...and I still don't see your point. The nature of how such communication works and ought be treated is the whole point of this discussion, so your first three lines don't convey anything to me. The fourth seems like it should clarify, but I don't get what you might mean. As for the fifth, referencing the example that I didn't follow isn't going to clarify the exact same example that I didn't follow.

From your fourth line, it sounds like you are agreeing with me, yet the way you responded to me sounds like you did not - so, I still don't understand what you are trying to say and how that relates to my post.
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Re: Assumptions in Math (Calculus) word problems

Postby z4lis » Sat Aug 23, 2014 3:52 pm UTC

@amad27

There's an enormous difference in how I write mathematics when communicating to my peers about my ideas and when communicating what I want students to do on an exam. When I am speaking to a friend or professor, I am almost never testing their knowledge of a topic. It makes no sense to hide information from them when my purpose is to clearly and effective transfer some idea. The standard operating procedure for communication is in place for those conversations. On the other hand, when asking students what to do on an exam, I may hide information if the ability to recognize that information is missing or incomplete is being tested. If I'm a good teacher, my lectures should bring up the fact that the problems may be missing information, and the test will see if you can understand how to perform with missing or extraneous information, and I might even be kind enough to alert the students to the "trick question". I would say that my lectures or subtle hints provide the context to signify some nonstandard conversation assumptions are in place, as opposed to some sort of change in tone, laughter, facial expressions, etc.
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Re: Assumptions in Math (Calculus) word problems

Postby Amad27 » Sat Aug 23, 2014 4:27 pm UTC

z4lis wrote:@amad27

There's an enormous difference in how I write mathematics when communicating to my peers about my ideas and when communicating what I want students to do on an exam. When I am speaking to a friend or professor, I am almost never testing their knowledge of a topic. It makes no sense to hide information from them when my purpose is to clearly and effective transfer some idea. The standard operating procedure for communication is in place for those conversations. On the other hand, when asking students what to do on an exam, I may hide information if the ability to recognize that information is missing or incomplete is being tested. If I'm a good teacher, my lectures should bring up the fact that the problems may be missing information, and the test will see if you can understand how to perform with missing or extraneous information, and I might even be kind enough to alert the students to the "trick question". I would say that my lectures or subtle hints provide the context to signify some nonstandard conversation assumptions are in place, as opposed to some sort of change in tone, laughter, facial expressions, etc.


Very true @z4lis.

But then again, how are we supposed to interpret problems?

Since every problem is not solvable, you cant say you must interpret a problem so that it is solvable.

Then how do you interpret word problems in the context of assumptions and what exists?

I suppose you are right that it makes no sense to hide information. But is that really the reason?

Forest Goose wrote:
Amad27 wrote:Hello,

You didn't understand the point of my last post.

The point is math requires communication.

The point of "requires communication, is that you must use communication skills, while interpreting the word problems.

The "interpreting the word problems" means when something isnt stated, it doesnt exist.

Like the example of the who I invited to my "party."


Why are you bolding things at me, the random bolded phrases don't really convey any extra information, they just look kooky.

...and I still don't see your point. The nature of how such communication works and ought be treated is the whole point of this discussion, so your first three lines don't convey anything to me. The fourth seems like it should clarify, but I don't get what you might mean. As for the fifth, referencing the example that I didn't follow isn't going to clarify the exact same example that I didn't follow.

From your fourth line, it sounds like you are agreeing with me, yet the way you responded to me sounds like you did not - so, I still don't understand what you are trying to say and how that relates to my post.


Sorry, I bolded so that it would "stick out" to you that I meant it was an "important fact."

Yes, the fourth line is true.

But I am trying to find the reason behind that. =)

I mean, it isn't an axiom, is it?

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Re: Assumptions in Math (Calculus) word problems

Postby Tirian » Sat Aug 23, 2014 5:56 pm UTC

Forest Goose wrote:And, at which point do we not put up with people treating problems as if they were "real world"? If I make a test that says give me an example of a turing machine that recognizes the primes, would it be a legitimate answer for a student to write, "Use the C++ library X instead.". That sounds stupid, at first, but it is far more practical, far more reasonable, and far more like what you would actually do; yet, I'm pretty sure no one would accept that as a legit answer to the problem. So why should people worry over how much a balloon can inflate?


I don't think I communicated my point well with my balloon problem analogy, so I'll switch to your metaphor.

When I learned to program in high school, we were learning how to design and implement structures like lists and queues and all, with all the basic search and sort algorithms. When I actually found my way into the field as a software engineer, STL had packaged all of those algorithms into tested and optimized libraries. I'd be foolish to apply my education to my coding problems. Should we have been teaching students in that era how to use the STL instead of how to reinvent the wheel themselves? Absolutely! That's not just more relevant, but it saves time and frustration from writing and debugging low-level code. With that free time, I'd hope that they had could study how to choose the appropriate data structures for given problems. They should be able to answer a problem like this on their final, in my opinion:

"You're on a team that is designing a software framework for a large call center. You are going to manipulate a collection of records representing incoming callers and another collection of records representing operators who will answer the calls. Choose an appropriate data structure for the collection of callers, and justify your decision. What are the drawbacks of your choice, and how big of a deal is it?"

That's not really a trick question -- it's the sort of thing that software engineers will contemplate all the time in the field. Answering it properly requires that you understand both data structures and what factors in real-world problems would influence the design of the software? Yes, you'd have more predictability in the test question if you added in that callers should be handled in the order their calls are placed and that people who hang up while on hold need to be removed from the structure somehow. But there's something wrong with training a software engineer with the expectation that those observations will be made for you.

Finally, a mathematics course should teach mathematics, I'm not sure why it should be teaching critical thinking. If a student hasn't learned any critical thinking by that point, tricky calc. questions aren't going to teach it to them


This reminds me of a chemistry teacher who was frustrated that the English department wasn't turning out students who knew how to write lab reports and that he wasn't going to bother with anyone who had been left so far behind. Even though he was the first exposure students had ever had to formal chemistry and he was uniquely qualified to appreciate the nuances of this very specific form of writing, he had managed to convince himself that teaching writing was someone else's problem.

The same thing goes triple for mathematics. We are the field of study that thinks about solving problems. All we do is talk about the strategies that Euclid and Descartes and Euler came up with for solving problems. We do it so that our students will be able to solve their own problems in their careers and lives, which may or may not be related to Euler's problems. If a math teacher has the attitude that they will teach their students to memorize and apply a dozen different formulas but their job is not to teach their students how to think more efficiently, they they will be doomed to a long and frustrating career of failing their students.

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Re: Assumptions in Math (Calculus) word problems

Postby Amad27 » Sat Aug 23, 2014 6:08 pm UTC

Tirian wrote:
Forest Goose wrote:And, at which point do we not put up with people treating problems as if they were "real world"? If I make a test that says give me an example of a turing machine that recognizes the primes, would it be a legitimate answer for a student to write, "Use the C++ library X instead.". That sounds stupid, at first, but it is far more practical, far more reasonable, and far more like what you would actually do; yet, I'm pretty sure no one would accept that as a legit answer to the problem. So why should people worry over how much a balloon can inflate?


I don't think I communicated my point well with my balloon problem analogy, so I'll switch to your metaphor.

When I learned to program in high school, we were learning how to design and implement structures like lists and queues and all, with all the basic search and sort algorithms. When I actually found my way into the field as a software engineer, STL had packaged all of those algorithms into tested and optimized libraries. I'd be foolish to apply my education to my coding problems. Should we have been teaching students in that era how to use the STL instead of how to reinvent the wheel themselves? Absolutely! That's not just more relevant, but it saves time and frustration from writing and debugging low-level code. With that free time, I'd hope that they had could study how to choose the appropriate data structures for given problems. They should be able to answer a problem like this on their final, in my opinion:

"You're on a team that is designing a software framework for a large call center. You are going to manipulate a collection of records representing incoming callers and another collection of records representing operators who will answer the calls. Choose an appropriate data structure for the collection of callers, and justify your decision. What are the drawbacks of your choice, and how big of a deal is it?"

That's not really a trick question -- it's the sort of thing that software engineers will contemplate all the time in the field. Answering it properly requires that you understand both data structures and what factors in real-world problems would influence the design of the software? Yes, you'd have more predictability in the test question if you added in that callers should be handled in the order their calls are placed and that people who hang up while on hold need to be removed from the structure somehow. But there's something wrong with training a software engineer with the expectation that those observations will be made for you.

Finally, a mathematics course should teach mathematics, I'm not sure why it should be teaching critical thinking. If a student hasn't learned any critical thinking by that point, tricky calc. questions aren't going to teach it to them


This reminds me of a chemistry teacher who was frustrated that the English department wasn't turning out students who knew how to write lab reports and that he wasn't going to bother with anyone who had been left so far behind. Even though he was the first exposure students had ever had to formal chemistry and he was uniquely qualified to appreciate the nuances of this very specific form of writing, he had managed to convince himself that teaching writing was someone else's problem.

The same thing goes triple for mathematics. We are the field of study that thinks about solving problems. All we do is talk about the strategies that Euclid and Descartes and Euler came up with for solving problems. We do it so that our students will be able to solve their own problems in their careers and lives, which may or may not be related to Euler's problems. If a math teacher has the attitude that they will teach their students to memorize and apply a dozen different formulas but their job is not to teach their students how to think more efficiently, they they will be doomed to a long and frustrating career of failing their students.



Its' funny because I never thought about this whole issue about assumptions before. I stumbled on a problem, and that got me confused.

I don't know how "assuming there might be a hole" is thinking more efficiently to be honest.

But what the person said about "finally, a mathematics course should teach mathematics, I'm not sure why it should be teaching critical thinking"

has made a good point. Even though I posted the question, it does feel stupid when you are critically thinking about the hole in the balloon. That's not the point, the point is to teach, not learn the anatomy of the balloon or trough or whatever.

I think the Chemistry teacher may have been out of his senses, but to an extent he is correct. It is the job of the English department to teach how to write formal or informal papers. Not his job. His job is to TEACH Chemistry. He isn't an English teacher.

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Re: Assumptions in Math (Calculus) word problems

Postby somehow » Sat Aug 23, 2014 7:46 pm UTC

Tirian wrote:
Forest Goose wrote:And, at which point do we not put up with people treating problems as if they were "real world"? If I make a test that says give me an example of a turing machine that recognizes the primes, would it be a legitimate answer for a student to write, "Use the C++ library X instead.". That sounds stupid, at first, but it is far more practical, far more reasonable, and far more like what you would actually do; yet, I'm pretty sure no one would accept that as a legit answer to the problem. So why should people worry over how much a balloon can inflate?


I don't think I communicated my point well with my balloon problem analogy, so I'll switch to your metaphor.

When I learned to program in high school, we were learning how to design and implement structures like lists and queues and all, with all the basic search and sort algorithms. When I actually found my way into the field as a software engineer, STL had packaged all of those algorithms into tested and optimized libraries. I'd be foolish to apply my education to my coding problems. Should we have been teaching students in that era how to use the STL instead of how to reinvent the wheel themselves? Absolutely! That's not just more relevant, but it saves time and frustration from writing and debugging low-level code. With that free time, I'd hope that they had could study how to choose the appropriate data structures for given problems.


It sounds like you're assuming that the point of an education in computer science is to prepare students to be competent software engineers. To me, that sounds sort of like claiming that the point of an education in mathematics is to prepare students to be competent accountants. You clearly don't think that that's the point of mathematics any more than I do: later in your post you describe math as "the field of study that thinks about solving problems", which I think I basically agree with. I wonder why we see computer science so differently.
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