The student is absolutely correct. I don't think it's even open for debate. Cutting anything in half requires exactly one cut; cutting in thirds requires two. It's as simple as that. The teacher that crafted the question, or worse yet, the publisher of a textbook that may have provided the test question, needs to take a hard look at whether or not they are in the correct profession.
The fact that the teacher not only marked the answer wrong (which could have just resulted from looking at a publisher-provided answer key) but actually wrote down a completely incorrect justification for the teacher's incorrect answer is rather disturbing to me. Also, this did not occur in a vacuum. Either no other students answered the question correctly, or the teacher saw the question being answered correctly by others and repeatedly marked it wrong with the same justification. Either way, it causes concern about the teacher.
You can't tell if it was a simple "whoops, I thought this question belonged to a problem category X, and I overlooked that it does not" typo-like mistake, meaning the teacher would instantly realize his/her mistake if you point it. Or if they wouldn't get it even after you try to explain it to them (what you're trying to imply here).
When grading things, ppl usually face hundreds of copies at a time and it's very tedious. It's easy to scrutinize a single highlighted problem that someone got wrong in hindsight, not realizing the person might've only dedicated 7 seconds to this problem out of 1000 others that were graded correctly.
I personally try to give them the benefit of doubt and assume best case scenario (but I also understand it might not be).
I agree with this. I think it's a pretty decent example of the fundamental attribution error.
The context of the problem clearly sets it up as one of those "everyone gets this wrong, so make sure you think a second" situations (I had to think a second, anyway).
I'd extend that to the publisher as well, though (or at least it's individual employees creating the book). First off, assuming the answer key has "15", I'm not in any way saying it's okay that we have text books teaching clearly incorrect information; I'm also not in the know on how 3rd grade math textbooks are created. That said, I've been in tons of jobs where you're expected to produce a crap ton of work at a breakneck pace and god help you if you want five minutes to check your work for dumb errors, because y'aint gettin it.
Of course, it could also easily be said that this is also the publishers fault for creating a working environment that isn't sufficiently rigorous or overburdens the employees.
On the topic of how math textbooks are created, you might like this commentary from Richard Feynman when he served on a school-math-textbook recommendation committee:
"In algorithm classes, there are often many right answers"
I accept this, but I'm assuming the original algorithm that a creative student produces will pass tests/produce same output as the 'textbook' solution. My understanding of the original article is that a correct final answer was marked wrong.
"Same goes for most college math."
Absolutely. My favourite from 16+ maths (GCSE in UK) is the area of a trapezium. Most find the mean length of the two parallel sides and multiply that by the distance between the parallel sides (so make a rectangle of the same area). About 1 in 15 break the trapezium up into two triangles and add the areas.
The third option would be to chop off both "wings" and consider it as a square plus two triangles. That is often my instinct when I forget to do it with the mean length of the parallels.
True, but I think that maybe 1/3 or more of the students would have answered this correctly. At some point during the grading process, you would think the teacher would begin to wonder why all of these students would have answered such a seemingly simple question "incorrectly" and take a second look.
It's not a trick question, just a bad one. It doesn't include enough information to give an informed answer. Telepathy is required to suss out the author's intention.
In the real world, you can ask more questions and get a more complete picture. On an exam, generally you must accept what you are given.
I'd be less worried if this seemed like a one-off thing (or if math professors were obligated to drive exclusively on bridges designed by their own students, heh).
As it is, this is one case among many (not all about grade schoolers and not all 'stories on the Internet' by a long shot) and the professor doesn't always acknowledge they were wrong. Speaking as an engineer, the work is hard enough when you do understand the math.
My god, this looks more like a 4chan troll post than stackexchange. I'm not convinced this really happened. Is this the only kid in the class that got it right? Did the teacher not then notice when the brighter kids were coming up with 20 min that there may be something to it, and reconsider the question himself/herself? So much fail in so little space. Ugh.
In high school geometry, I remember my teacher making some assertion that was plainly false - I think it was that 3 planes always intersect in a line. After arguing with him for like 15 minutes, I walked to the front of the class and wrote a proof on the board. I spent the rest of the class period sitting outside.
You're not convinced this actually happened? When I was in elementary school, I regularly (ie. several times per semester) got into arguments with my math and science teachers over stuff this dumb. There's no need to make up something like this when you can find it in the real world so easily.
In grade school I had an argument with my science teacher about wheels. She said that a point along the outside of a wheel moved faster than a point nearer to the center (which is absolutely correct). However, she followed that up by saying that the outside of the wheel makes more revolutions than the inside. I tried to correct her, but she wasn't having any of it. So I grabbed my bike from outside, brought it into the classroom and tied two pieces of string onto one of the spokes on the bike: one near the axel, one near the tire. A few spins of the wheel had her convinced, but I can't believe I actually had to do it.
If anything, the inside should have to make more revolutions. Since the whole bike is traveling at 10 MPH, at the point with the smaller radius, you'd need more revolutions to travel that same linear speed.
A fence is made using 15 posts spaced equally along a straight line. There are 3m between each post. What is the distance between the first and last post?
When I'm teaching this kind of thing, we go out and walk around the building site opposite with a few 15m measuring tapes. The physical walking out and measuring helps.
Change "There are 3m between each post." to "The distance from one post centre to the next is 3m"
Which illustrates the general point: you need teams working the test and checking their answers against what the writer thought the answers were. You also need English specialists checking the wording of the questions.
The question does not say cut "into thirds," it says "into three pieces." This - http://i.stack.imgur.com/kEjP0.png - is a perfectly reasonable answer which, assuming the rate of cutting is constant, would result in 15 minutes.
It's a bad question.
Edit: That said, I would have given the same answer as the student, because I think that's the most reasonable interpretation, especially considering the illustration. But the keyword there is "interpretation." The question is ambiguous.
You are correct, but you can trisect that piece of wood an infinite number of ways. The logical equilibrium point is 3 even pieces.
The student chose a ratio of 1,1,1; which is the logical equilibrium point. your image shows 1.5,0.75,0.75; which is the second most logical ratio because it is in the form x + 2y = 3 (which can be trisected an infinite number of ways while maintaining that ratio). The third form would be x + y + z = 3; which can also be trisected an infinite number of ways and would be the least intuitive.
i am agreeing with you, i am just trying to show that it is illogical for it to be 'open for debate'.
There is a game theory term for this type of equilibrium, but i forgot its name. Its the same type of equilibrium as "there are three colors and a number, which one is different?" type sesame street problems.
I didn't see the picture at first, and reasoned just as you exposed.
However, the teacher corrects it by writing "4 = 20". This is plainly wrong and with no possible explanation, since following the above reasoning, cutting in 4 pieces would require: 10 + 10 / 2 + (10 / 2) / 2 = 17.5 minutes.
I can cut a piece of wood into thousands of pieces in 5 seconds. I just slice it across the top a few times with my saw, and all the sawdust that comes off counts as separate pieces.
As someone said below, it's only open for debate if you want to be pedantic. The Dr. Sheldon Cooper's among us may debate it, but it's pretty obvious what the question was looking for. There is even an illustration showing the cut, which would take an identical amount of time.
But even with your picture the answer can be 20 seconds. You're assuming the person is starting at the top of the line and cutting all the way through the board to the bottom. But they could just as easily rotate the board 90° and cut across a different axis. Assuming a 1" thick board, this means they're cutting through 1" of wood on each cut, meaning both cuts take the same amount of time.
this was my first thought, but then I realized the teacher gave justification for his answer.
the problem is poorly formulated. The teacher would have been correct if it had said "it took 10 minutes to cut away 2 pieces from a very large board (thus resulting in 2 cuts, 3 pieces total)", whereas the student's answer assumes a single cut, which is more reasonable.
Most likely, whoever produces and publishes the question sets changed the question for a new edition and did not notice that the new question also changed the answer.
In the previous edition it was probably something like "Marie works in a factory which makes cars; it takes her 10 minutes to finish two cars. How long will it take Marie to finish three cars?"
And the answer to that would be 15 minutes, and the reasoning in the answer (based on reducing fractions, which is what it's probably supposed to teach) would be correct.
But probably in the next edition the question changed from putting things together to cutting them apart, and the author/editor simply didn't realize that these are not interchangeable. The teacher, meanwhile, probably didn't look too closely at it, and simply applied the answer and reasoning supplied in the teaching materials for the question set.
None of which implies that the teacher can't do the math; rather, it implies systemic problems in the way the materials are produced and in the methods used by teachers to grade the work.
While, given the picture presented the child's solution makes the most sense, there is another close scenario in which the teacher is correct[1]. So its not really "as simple as that."
this is not a plausible scenario at all. if the quantity of work done is not equal at each step the question is impossible to answer which means that the teacher is nor right, just there is a possibility to be right. but then, every answer could be right, just adjust your cutting path according to the answer you would like to be correct.
Yes, the question did not account for unknown specifics. There is also a 3rd answer which in which the answer accounts for the person making the cuts having to answer the phone half way through the job.
Context is everything, this is a question for a 3rd grader not someone who is in Calculus. I think it's safe to assume the student is right given the screenshot.
It is ABSOLUTELY open for debate, and part of the clue is in the question "if she works just as fast" ie. the cutting rate is constant. Then, it is ambiguous since the SIZE of the pieces is not mentioned.
It's not the teacher's fault, per se; the question is unanswerable. The student picked one interpretation but the (likely) correct one is shown in the answer http://math.stackexchange.com/a/380007
It's only open for debate if you're being extremely pedantic. There is even an illustration demonstrating exactly what the cuts look like!
Yes, if you want to be very nit-picky the question is undefined, but this is a third grade math test and the only reasonable answer is 20min. The student was absolutely correct.
I agree. If I am the principal and the student protests to me and the teacher gives some bizarre-ass interpretation of the problem to me, the teacher will have a problem with me.
I've seen enough of that in high-school physics where the teacher literally doesn't understand what the hell the problem is asking to piss off the Good Humor man.
There is a picture of that illustrates the cut that is being made well enough to infer.
Some inferences have to be made, this is a human taking the test not a robot, and it's a 3rd grade test.
Even if the size of the pieces were specified, she could be cutting a different kind of wood or using a different saw or the humidity level could be different, but she's still "working just as fast".
3rd graders would be confused if you attempted to be completely unambiguous with this time of question.
Look at the image provided with the question. It shows a saw cutting the board in a way that leaves it unambiguous - the cutting time for that cut would be identical regardless of where each cut took place.
In reality, it probably was not open for debate. This is third grade math, remember. The student probably spent the last three problems working similar questions covering ratios/fractions. And the student probably spent at least a few weeks of school working similar types of problems on homework. After so long taking these types of classes and tests, if a student answers a question without using knowledge gained from the class and gets the question wrong, well...what did they expect?
(Not saying I like it, but it's the way it is many places.)
I would assume that the "*" next to the problem was an indication that this what not a 'time to fill a water bucket' problem and that more thinking would be required than the previous 3 problems.
If it is ambiguous, there is no answer. There must be an answer. Therefore, it cannot be ambiguous.
The answer given is the only one it is possible to give. Therefore, it must be the correct one.
The context isn't so much "third grade" as it is "math test", and very, very few math tests allow "Question ill-formed as posed" as a valid answer. Maybe more should.
That's actually a great idea. If I were a math teacher, I would teach my class that IFQ is a reasonable answer to a question, and I'd throw in a few plainly ill-formed questions just to keep them on their toes. Actual thinking > correct answers.
I think for most questions that are not very straight forward, IFQ would be a valid answer with only very little argumentation. That's why formal languages are needed.
The first round of the UK Maths Challenge * is multiple choice, and does often include questions with "not enough information provided" as one of the available answers. However, this isn't a mechanism for identifying badly phrased questions.
* (the feeder competition for the British Mathematics Olympiad, and then the International one)
The fact that the teacher not only marked the answer wrong (which could have just resulted from looking at a publisher-provided answer key) but actually wrote down a completely incorrect justification for the teacher's incorrect answer is rather disturbing to me. Also, this did not occur in a vacuum. Either no other students answered the question correctly, or the teacher saw the question being answered correctly by others and repeatedly marked it wrong with the same justification. Either way, it causes concern about the teacher.