The story is a wonderful illustration that the human brain is not perfect. It seems that most people when first reading the math problem get it wrong. Our brain is designed to first jump to conclusions before seriously thinking about the problem. The human mind may be the highest form of intelligence on the planet, but that does not mean that there are not serious design flaws. The human brain was born out of a process of Evolution, and is designed to function in a natural setting. Perhaps in a distant future, when humanity has created true A.I., it will be possible to observe just how biased and illogical the human mind really is by comparing it to artificial intelligence.
The problem is not if the humar brain is/isn't perfect (compared to what?).
The matter here is that the question is not mathematically strict and so the reader is free to interpret it as he pleases, and multiple solutions spawns naturally.
The teacher is very mistaken trying to assert a unique solution.
It's more a logic question than a math one. The confusion spawns from the fact that the three numbers present in the question are 10, 2, and 3 (so the thought process would be 2 = 10 min so 1 = 5 min, thus 3 = 15 min).
But 2 represents the final state, though requires only 1 action (cut). And the required answer (time spent) is related to the number of actions, not the final state.
This reminds me of the water lily problem: a water lily doubles in size every day. It takes 30 days to cover the whole pond. How many days does it take for the water lily to cover half the pond? (Answer: 29, not 15).
I hadn't heard of the water lily problem either. The answer was completely obvious to me, but I don't think this was because I'm particularly intelligent or math savvy, but rather because I'm used to working in binary, which I think gives you a better intuitive sense of the concept of doubling.
My teacher used to give me this example when I was a kid: You see one matchstick with one eye. How many will you see with two eyes? :-)
Here's another one that used to confuse high school students in my class: You look at a 10 degrees angle with a lens of 3X magnification. How much would the angle look like? :-)
the student is right because it states "into 2 pieces" which means you do one cut to an object and you now have 2 objects. this is total number of pieces = number of cuts + 1 from the beginning.
probably the person who graded the question assumed that you are cutting chunks from an object, like slicing a bread. for every cut(except the last one) you get one new object, so every cut is +1 new object. if you slice the whole thing and the remaining object can be +1 piece, just like in the first situation, if you consider the last piece equal to the pieces you cut.
Or clear communication and understanding of the specifications?
Seems impossible for anyone to interpret it differently than the student did, but from the comments it's clearly easy for people to extract ambiguity from what appears to be a simple, straightforward specification.
I would've arrived at the teacher's solution, but the question allows different interpretations and both answers are correct assuming different interpretations.
The correct answer would be "I do not know, this problem is under-specified."
Can you explain why you think it has two correct interpretations?
I obviously thought 15 min when I first read it and my brain didn't want to accept any other solution until I read the post below where it said 20 min and explained it as 2 pieces = 1 cut = 10 min, 3 pieces = 2 cuts = 20 min.
And now I can't see why my first thought was correct. Did you come up with some good rationale as to why it should be 15 min or other?
If you were to cut off two pieces of the board from an unknown source, you'd require two cuts. Three pieces would require three cuts (with the rest remaining behind). I don't think the wording really allows for this interpretation, but that is the only way I could explain the alternative answer.
>Can you explain why you think it has two correct interpretations?
Because it depends on whether you 1) require that the N pieces be congruent and 2) what counts as a cut. I think the textbook answer is based on assuming 1) no, and 2) cutting along a line segment at least as long as a side.
Alternately, what counts as a "board" and a "cut".
Then you get the answer by assuming you cut a square board in half, then one of the pieces into squares (which requires cutting along a line segment half as long).
Some people are arguing about whether 'cut into two' might really mean 'cut two off'. So I think your answer makes three interpretations (and nicely demonstrates that you do indeed have to specify things like "cutting is abstract and all cuts are of the same length").
The problem does specify 'works as fast' without any regard for length, though. And it obviously isn't actually a geometry problem because it doesn't even specify any ratios or angles - you could just cut a corner off and be done in a few seconds!
There's a picture next to the question of a saw cutting a straight thin plank. Its very unlikely that this interpretation of the question was intended.
I agree with you here. If a board is 2 x 2 and you cut it in half, you have 2 1x2 pieces. That took 10 minutes. If you cut one of those 1x2 pieces in half you could have 2 1x1 pieces of board, and one 1x2 piece of board, thus 3 pieces. The first cut took 10 minutes, the 2nd cut, being half the size of the original took 5 minutes, thus 15 minutes.
> The correct answer would be "I do not know, this problem is under-specified."
In that case, I'd write my assumptions about the problem and show how I arrived at the solution. If the teacher still says it's wrong, I probably won't bother arguing - I don't waste my time arguing with morons.
>but the question allows different interpretations and both answers are correct assuming different interpretations.
There is only one logical interpretation (though you'd have to actually think past the conclusion your brain jumps to), the question was perfectly clear about the board being cut into two pieces.
I disagree with your logic. As the corner-cut solution shows, there are only two sane answers. "20 minutes" where you have cuts across the plank or "any* amount of time whatsoever" where you allow any kind of cut.
There is no sufficiently logical way to get to any particular number other than 20; the shape of the plank does not allow you to cut across and make your cuts intersect like you might with a square board. There is no reason on this particular shape to prefer "15 minutes" over "14 minutes" or "25 minutes". It all gets lumped into "any amount of time whatsoever".
If "any amount of time whatsoever" was an acceptable answer it wouldn't make sense that a single cut takes 10 minutes, so we should discard that answer. This leaves only one candidate answer, 20 minutes.
*"any" would be limited by how long of a diagonal you can make but it would be hours
Oh the "simple" questions... reminds me of the fragment from Cryptonomicon where Lawrence Waterhouse answering the usual trivial math question about boat going from A to B with some speed X while the water moves with speed Y. He failed, even though he decided the answer cannot be that trivial and wrote a long solution involving analysing the flow of the water using partial differential equations (later published in a paper).
The problem at hand is "what are you supposed to do" vs the actual problem at hand.
At first I had a difficulty seeing why 20 should be wrong, but then it dawned upon me: The teacher set out to create a word problem for a specific mathematic solution strategy. Students probably were inundated with this strategy for weeks before the test, so for them it is very clear what they were supposed to do.
Absolutely. The test was probably written by the person grading it to cover fraction/ratio problems. The question shown is a poor rewrite of something like, "If it takes 10 minutes to fill two buckets, how long does it take to fill three buckets?"
The format of the page looks like something out of a book, not something that the teacher made. My guess of what happened is that when the teacher went to solve the problem, she read it as the bucket problem.
Thanks, finally I see the reason behind the teacher's solution...
I think, this is a good example why you should not divide math problems in rigid cetegories. Things become worse when badly taught high school students go to college, and fail to do simple arithmetics and algebra.
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).
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 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.
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.
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
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.
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.
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)
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.
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.
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.
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."
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.
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.
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.
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.
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.
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.
I'm impressed that the student thought it through, but people are giving the grader too much of a hard time. If the question was instead, "If a machine can produce 2 cars in 10 minutes, how long does it take to produce 3 cars?" the teacher would be correct. If you've ever taken a standardized math test, it's easy to assume that the question is just a variation of that classic question. If I were a third-grader, I would have probably answered "???". So kudos to this kid.
> "If a machine can produce 2 cars in 10 minutes, how long does it take to produce 3 cars?"
This question is also ambiguous, because there is no info about how long the operation takes, e.g. the machine may be parallelized and produce a 3rd car in 10 minutes along with the 2 others or that the machine may obey a non-linear increase in production time per unit.
A friend of mine teaches school in rural North Carolina - here's what she tells me.
Her school has to meet certain percentage-based "standards" - I forget the exact numbers, but let's say 75% is the cutoff. So now when Joey gets 5 answers right out of 10, the resulting 5/10 is defined as "75%."
In my first semester (I think, might have been second) honors calculus class, the (great) teacher got carried away on one midterm. I got something like 40%, and that was the second highest grade in the class, the average was more like 30%. He was so disappointed we didn't do better on that exam...
Curve grading is just conceptually silly. If a test does indeed cover the material, then answering correctly half of it should lead to a 50%, no more, no less.
If curve grading is required because the test doesn't properly assess what the students have been working on, that means the test was bad in the first place.
My undergrad professor in electrical engineering asked us to calculate voltage on a diode as part A of a problem. This came out as 0.7V. The part B was to convert this to percentage! (Don't ask me how and why the answer was 70%).
>Her school has to meet certain percentage-based "standards"
I went to a top-5 public high school in my state. "Standards" are so ridiculously low it's hilarious. I'm pretty sure you could still exceed the state standard for 12th grade reading with the reading level most of us (upper middle class, white, college-educated parents, high property taxes) had in 5th grade. Meeting standards certainly didn't mean you were even remotely qualified to go to college, and is orders of magnitude below the aptitude required to get into good colleges. So when I hear about districts where just reaching the standards is a stretch, it's shocking just how enormous the gulf in education quality in this country really is.
IMO this is a great argument to stop controlling schools at such a hyper-local level. There's no good reason for K12 education to vary geographically. The education that the professional world will expect of a kid in Chicago is the same as what it will expect from a kid in small-town Alabama or rural North Carolina. Why do we accept the argument that K12 education should be up to the community? Why is preparing workers for a global economy considered a local problem?
Because it's disgusting just how better-prepared I am than the children in your friend's school district. I didn't earn parents who can afford to live in an expensive community, I didn't earn the ability to take AP classes from talented teachers, I didn't earn a calculus teacher who refuses the school-provided textbooks in favor of illicit PDFs from a curriculum being drafted by one of her colleagues, I didn't earn a veteran teacher and former DuPont research scientist to get me a 5 in AP Chem. All our STEM AP programs get 4s and 5s save for a small handful of slackers; the teachers calm us down when we're getting nervous by reminding us that we're being graded on a curve alongside kids from the middle of nowhere. The opportunties we had that other communities don't is just staggering.
This is a classical question I ask to children (and I was asked as a child too). It was/is fun, because it is easier to answer if you haven't yet started arithmetic, or if you can manage to step outside the pressure of this new thing that you are being taught at school.
How many cuts do you need to make in order to split a board into 2?
How about 3?
How about 4?
In this case, the teacher has failed. But, everybody must have learned something out of this.
The answer to this question is open for debate. You see you didn't specify whether you cut all the way through resulting in two halves of a person with one cut on one of them. And which one!
If you ask me to make these cuts with a chop saw (or mitre saw if you prefer) I am making 1, 2 and 2 cuts. Where one cut is defined as pulling the trigger on the saw arm and pressing the handle down.
As I stated downthread: If you ask me to make these cuts I am plugging in my chop saw (or mitre saw if you prefer) I am making 1, 2 and 2 cuts. Where one cut is defined as pulling the trigger on the saw arm and pressing the handle down.
Your "vice" will be my left hand pushing the wood against the fence and towards the stop block. Do you do a lot of woodworking?
It was not snark. I was just trying to ascertain if you were unfamiliar with working with lumber or just being difficult. Would you really make three cuts?
For anyone else initially as confused as I was, dfc is saying that after the first cut, they would stack the resulting two pieces for the 2nd cut with the chop saw.
> But, everybody must have learned something out of this.
Let's hope the lesson learnt is not "math is too hard for me; I'm stupid; I don't understand this; I tried to ask my teacher but they're authoritarian and because I'm just a kid I don't know the socially acceptable way to ask this kind of stuff and the teacher got all defensive and punished me, and so I must never question anyone, even when I think I can show that I'm right and I think they've made a mistake".
Teaching is a hard job. Many parents don't support you at all. It's politicised (at least, in England it's very political). It's low status. So, I'm not really knocking the teacher. I do hope that after a chat the teacher gave the child better marks.
Anybody talking about how this problem is ambiguous or under-specified are of course technically correct. By making that claim though, you are ignoring the context of the problem!
This is a 3rd grade math test that even includes an illustration of how the cuts are made! Within that context, the answer is unambiguously 20 minutes.
Not sure about that under-specified... adding more info/clarification would mean simply giving the answer. Clicking that link i was expecting something unintuitive, but that was not the case, i agree, it's really just a 3rd grade problem.
Some kids are smarter than others in a given grade, and would spot the ambiguities nevertheless. This used to be a problem I consistently faced in both school and college.
I have to laugh at how much play this got at Stack Exchange and here. This is simple, scare quotes are unnecessary. The teacher made a mistake. They're not perfect, they make mistakes just like the rest of us. End of story.
It's quite scary though, at least to me. From the scribbles the teacher made, it looks like they are being taught to deal with fractions in the most mechanical way possible.
It's quite scary though, at least to me. From the scribbles the teacher made, it looks like they are being taught to deal with fractions in the most mechanical way possible.
This question is easy in hindsight. The fact that it's been prefaced as something "simple" makes you scrutinize it much more closely than if you were someone grading a series of questions en masse...because you've been warned that it's not so simple.
That said, this gave me a little glimmer of hope about the state of logic education, at least among our third grade students.
I felt a great disturbance in the math as if a million minds applied themselves to a problem and were suddenly silenced. I fear something terrible has happened.
Perhaps the teacher or the author of the question understood the problem differently – we are cutting off small pieces from a long stick. So to cut off 2 pieces, we need 2 cuts, not 1.
very clever. this ambiguity shows how difficult it is for creative students and how important it is to be graded on the process as well as the conclusion. The process would show your divergent thinking and why you arrived at a different answer.
173 comments
[ 2.4 ms ] story [ 190 ms ] threadThe matter here is that the question is not mathematically strict and so the reader is free to interpret it as he pleases, and multiple solutions spawns naturally.
The teacher is very mistaken trying to assert a unique solution.
But 2 represents the final state, though requires only 1 action (cut). And the required answer (time spent) is related to the number of actions, not the final state.
This reminds me of the water lily problem: a water lily doubles in size every day. It takes 30 days to cover the whole pond. How many days does it take for the water lily to cover half the pond? (Answer: 29, not 15).
Here's another one that used to confuse high school students in my class: You look at a 10 degrees angle with a lens of 3X magnification. How much would the angle look like? :-)
probably the person who graded the question assumed that you are cutting chunks from an object, like slicing a bread. for every cut(except the last one) you get one new object, so every cut is +1 new object. if you slice the whole thing and the remaining object can be +1 piece, just like in the first situation, if you consider the last piece equal to the pieces you cut.
so, +1 to the student :)
Seems impossible for anyone to interpret it differently than the student did, but from the comments it's clearly easy for people to extract ambiguity from what appears to be a simple, straightforward specification.
Some people list off-by-one errors as the third hardest thing.
0) cache invalidation
1) naming things
2) off by one errors
Looks like he counted right to me.
EDIT: fixed newlines
Looks like he counted right to me.
The correct answer would be "I do not know, this problem is under-specified."
I obviously thought 15 min when I first read it and my brain didn't want to accept any other solution until I read the post below where it said 20 min and explained it as 2 pieces = 1 cut = 10 min, 3 pieces = 2 cuts = 20 min.
And now I can't see why my first thought was correct. Did you come up with some good rationale as to why it should be 15 min or other?
Because it depends on whether you 1) require that the N pieces be congruent and 2) what counts as a cut. I think the textbook answer is based on assuming 1) no, and 2) cutting along a line segment at least as long as a side.
Alternately, what counts as a "board" and a "cut".
Then you get the answer by assuming you cut a square board in half, then one of the pieces into squares (which requires cutting along a line segment half as long).
The problem does specify 'works as fast' without any regard for length, though. And it obviously isn't actually a geometry problem because it doesn't even specify any ratios or angles - you could just cut a corner off and be done in a few seconds!
If it was dressed, that 1 might be very small indeed.
In that case, I'd write my assumptions about the problem and show how I arrived at the solution. If the teacher still says it's wrong, I probably won't bother arguing - I don't waste my time arguing with morons.
There is only one logical interpretation (though you'd have to actually think past the conclusion your brain jumps to), the question was perfectly clear about the board being cut into two pieces.
There is no sufficiently logical way to get to any particular number other than 20; the shape of the plank does not allow you to cut across and make your cuts intersect like you might with a square board. There is no reason on this particular shape to prefer "15 minutes" over "14 minutes" or "25 minutes". It all gets lumped into "any amount of time whatsoever".
If "any amount of time whatsoever" was an acceptable answer it wouldn't make sense that a single cut takes 10 minutes, so we should discard that answer. This leaves only one candidate answer, 20 minutes.
*"any" would be limited by how long of a diagonal you can make but it would be hours
At first I had a difficulty seeing why 20 should be wrong, but then it dawned upon me: The teacher set out to create a word problem for a specific mathematic solution strategy. Students probably were inundated with this strategy for weeks before the test, so for them it is very clear what they were supposed to do.
I think, this is a good example why you should not divide math problems in rigid cetegories. Things become worse when badly taught high school students go to college, and fail to do simple arithmetics and algebra.
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.
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).
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.
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.
http://www.textbookleague.org/103feyn.htm
Oh, yes, been there and I have the video. Don't you work from pre-written and checked marking schemes?
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.
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.
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
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'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.
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.
(Not saying I like it, but it's the way it is many places.)
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.
* (the feeder competition for the British Mathematics Olympiad, and then the International one)
;-)
[1]http://math.stackexchange.com/a/380007
Here is another version:
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.
I've also had students in Functional Maths classes just sketching arrangements of posts and counting the spaces.
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.
Yeah, my hand is firmly down too…
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.
(My argument is taken from this answer: http://math.stackexchange.com/a/380007 )
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.
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.
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.
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.
Kidding aside, this is probably a good demonstration of how shoe stringing our education budgets might not be the best idea.
This question is also ambiguous, because there is no info about how long the operation takes, e.g. the machine may be parallelized and produce a 3rd car in 10 minutes along with the 2 others or that the machine may obey a non-linear increase in production time per unit.
Her school has to meet certain percentage-based "standards" - I forget the exact numbers, but let's say 75% is the cutoff. So now when Joey gets 5 answers right out of 10, the resulting 5/10 is defined as "75%."
We're doomed.
The prof would make the test very hard so the average was around 50-70 and then use a curve to get grades.
If curve grading is required because the test doesn't properly assess what the students have been working on, that means the test was bad in the first place.
What we're talking about here is remapping a fraction (what the student scored) to a higher-than-equivalent percentage (what the "standard" requires).
I went to a top-5 public high school in my state. "Standards" are so ridiculously low it's hilarious. I'm pretty sure you could still exceed the state standard for 12th grade reading with the reading level most of us (upper middle class, white, college-educated parents, high property taxes) had in 5th grade. Meeting standards certainly didn't mean you were even remotely qualified to go to college, and is orders of magnitude below the aptitude required to get into good colleges. So when I hear about districts where just reaching the standards is a stretch, it's shocking just how enormous the gulf in education quality in this country really is.
IMO this is a great argument to stop controlling schools at such a hyper-local level. There's no good reason for K12 education to vary geographically. The education that the professional world will expect of a kid in Chicago is the same as what it will expect from a kid in small-town Alabama or rural North Carolina. Why do we accept the argument that K12 education should be up to the community? Why is preparing workers for a global economy considered a local problem?
Because it's disgusting just how better-prepared I am than the children in your friend's school district. I didn't earn parents who can afford to live in an expensive community, I didn't earn the ability to take AP classes from talented teachers, I didn't earn a calculus teacher who refuses the school-provided textbooks in favor of illicit PDFs from a curriculum being drafted by one of her colleagues, I didn't earn a veteran teacher and former DuPont research scientist to get me a 5 in AP Chem. All our STEM AP programs get 4s and 5s save for a small handful of slackers; the teachers calm us down when we're getting nervous by reminding us that we're being graded on a curve alongside kids from the middle of nowhere. The opportunties we had that other communities don't is just staggering.
How many cuts do you need to make in order to split a board into 2? How about 3? How about 4?
In this case, the teacher has failed. But, everybody must have learned something out of this.
Your "vice" will be my left hand pushing the wood against the fence and towards the stop block. Do you do a lot of woodworking?
Let's hope the lesson learnt is not "math is too hard for me; I'm stupid; I don't understand this; I tried to ask my teacher but they're authoritarian and because I'm just a kid I don't know the socially acceptable way to ask this kind of stuff and the teacher got all defensive and punished me, and so I must never question anyone, even when I think I can show that I'm right and I think they've made a mistake".
Teaching is a hard job. Many parents don't support you at all. It's politicised (at least, in England it's very political). It's low status. So, I'm not really knocking the teacher. I do hope that after a chat the teacher gave the child better marks.
This is a 3rd grade math test that even includes an illustration of how the cuts are made! Within that context, the answer is unambiguously 20 minutes.
https://en.wikipedia.org/wiki/Parkinsons_law_of_triviality
That said, this gave me a little glimmer of hope about the state of logic education, at least among our third grade students.