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).
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.
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.
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.
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."
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 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).
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.
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 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.
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%."
I read all the comments on the math.stackexchange.com submission and all the comments here before starting to type this reply. There are a lot of issues here, and I will try to add the perspective of a mathematics teacher. The reason I can gain paying clients for my mathematics lessons even though I have no degree in mathematics and no degree in teaching is that I can produce results that many elementary school teachers in my market area cannot produce. Mathematician Patricia Kenschaft's article from the Notices of the American Mathematical Society "Racial Equity Requires Teaching Elementary School Teachers More Mathematics,"
reports on her work in teacher training programs for in-service teachers in New Jersey. "The understanding of the area of a rectangle and its relationship to multiplication underlies an understanding not only of the multiplication algorithm but also of the commutative law of multiplication, the distributive law, and the many more complicated area formulas. Yet in my first visit in 1986 to a K-6 elementary school, I discovered that not a single teacher knew how to find the area of a rectangle.
"In those innocent days, I thought that the teachers might be interested in the geometric interpretation of (x + y)^2. I drew a square with (x + y) on a side and showed the squares of size x^2 and y^2. Then I pointed to one of the remaining rectangles. 'What is the area of a rectangle that is x high and y wide?' I asked.
. . . .
"The teachers were very friendly people, and they know how frustrating it can be when no student answers a question. 'x plus y?' said two in the front simultaneously.
"'What?!!!' I said, horrified."
Professor Kenschaft's article includes other examples of the mathematical understanding of elementary schoolteachers in New Jersey. In this regard, New Jersey may actually set a higher standard than most states of the United States, so all over the United States, there is risk of learners being misled into incorrect mathematical conceptions by their schoolteachers.
The problem is not ideally written, to be sure. In February 2012, Annie Keeghan wrote a blog post, "Afraid of Your Child's Math Textbook? You Should Be,"
in which she described the current process publishers follow in the United States to produce new mathematics textbook. Low bids for writing, rushed deadlines, and no one with a strong mathematical background reviewing the books results in school textbooks that are not useful for learning mathematics.
But if you put a poorly written textbook into the hand of a poorly prepared teacher, you get bad results like that shown in the submission here. Those bad results go on for years. Poor teaching of fraction arithmetic in elementary schools has been a pet issue of mathematics education reformers in the United States for a long time. Professor Hung-hsi Wu of the University of California Berkeley has been writing about this issue for more than a decade.
he points out a problem of fraction addition from the federal National Assessment of Educational Progress (NAEP) survey project. On page 39 of his presentation handout (numbered in the .PDF of his lecture notes as page 38), he shows the fraction addition problem
12/13 + 7/8
for which eighth grade students were not even required to give a numerically exact answer, but only an estimate of the correct answer to the nearest natural number from five answer choices, which were
(a) 1
(b) 19
(c) 21
(d) I don't know
(e) 2
The statistics from the federal test revealed that for their best estimate of the sum of 12/13 + 7/8,
7 percent of eighth-graders chose answer choice a, that is 1;
28 percent of eighth-graders chose answer choice b, that is 19;
27 percent of eighth-graders chose answer choice c, that is 21;
14 percent of eighth-graders chose answer choice d, that is "I don't know";
while
24 percent of eighth-graders chose answer choice e, that is 2 (the best estimate of the sum).
I told Richard Rusczyk of the Art of Problem Solving about Professor Wu's document by email, and he later commented to me that Professor Wu "buried the lead" (underemphasized the most interesting point) in his lecture by not starting out the lecture with that shocking fact. Rusczyk commented that that basically means roughly three-fourths of American young people have no chance of success in a science or technology career with that weak an understanding of fraction arithmetic.
The way this is dealt with in other countries is to have specialist teachers of mathematics in elementary schools. Even with less formal higher education than United States teachers,
teachers in some countries can teach better because they develop "profound understanding of fundamental mathematics" and discuss with one another how to aid development of correct student understanding. The textbooks are also much better in some countries,
and the United States ought to do more to bring the best available textbooks (which in many cases are LESS expensive than current best-selling textbooks) into many more classrooms.
It seems likely that no one taught those students how to think about math.
I.e. teaching students the steps to solve a math problem is not teaching them how to think about the problem.
I instantly knew 12/13 + 7/8 was ~2 because I visualize two pie charts in my head, both of which are mostly full. This is in contrast to the other way to solve the problem, converting the fractions to a common denominator and then dividing by the denominator. It would take me some time to do the latter, whereas I can instantly do the former.
I don't think the students who got that wrong (nor some who got it right) do any kind of visualization in their heads.
Teachers need to realize that it's the operations in the head that count the most, not rote memorization of steps to solve a problem.
Educational books is something that really could work fantastically well with open source models. Some group of people prepare best current practice chapters for a single topic. This group includes educators (to know where children get confused and make mistakes) and experts (to spot subtle errors, and to 'foreshadow' knowledge needed later).
These are released.
People can make corrections.
For something like math this could have significant impact not just in the US and EU but in the developing world too.
PS: About the fraction multiple choice: There's probably a bad joke about 24% being what we'd expect if we let the students chose at random. I'm not funny enough to think what it is. (The punchline being that there are 5 options, not 4.)
It in no way detracts from your point, but I believe that asking a 'number sense' problem like the estimation or fraction problem you gave is a different issue than teaching the more algorithmic procedure of solving fraction problems.
One doesn't seem to preclude the other, nor does it seem to mean you won't have success in a science or technology career. I think you'll find a lot of people who know how to solve, say, 'circular motion problems,' but don't really understand what they are doing.
Just curious: which aspects of this elementary math education would you say could be taught by automated high-quality means, such as the combination of interactive games and questions that many people are working on right now? I assume that video lectures would not be effective at that age.
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.
> 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.
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.
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.
For me this question is more about careful reading that actual mathematics. A valuable lesson, IMHO.
As for the teacher, well, I and my entire class once spent half a lesson arguing with our maths teacher who was swearing blind that 1x1=2. She wasn't an idiot or any thing, actually usually a very good teacher, but she just had one of those silly mind blocks. Once it clicked in her head she basically realised how mad she looked and took it with great humour. So, fair enough. Only human.
This seems like a simple matter of too many authors. The spec called for a question of the form "it takes x minutes to do two things, how many does it take to do three?", the copywriter remembered vaguely some brain-teaser question from his pre-SAT prep book and wrote the text of that already having the answer chosen as x + x/2, and then the layout guy picked a nice saw cutting wood from his clipart CD.
Add it all up and it only takes third grade math to know it equals fail.
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.
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.
[+] [-] downandout|13 years ago|reply
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.
[+] [-] shurcooL|13 years ago|reply
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).
[+] [-] keithpeter|13 years ago|reply
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.
[+] [-] dhimes|13 years ago|reply
[+] [-] Total_Meltdown|13 years ago|reply
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 )
[+] [-] ubernostrum|13 years ago|reply
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.
[+] [-] chacham15|13 years ago|reply
[1]http://math.stackexchange.com/a/380007
[+] [-] jrabone|13 years ago|reply
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
[+] [-] bbx|13 years ago|reply
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).
[+] [-] jay_m|13 years ago|reply
[+] [-] randlet|13 years ago|reply
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.
[+] [-] drtse4|13 years ago|reply
[+] [-] alok-g|13 years ago|reply
[+] [-] driverdan|13 years ago|reply
[+] [-] jdiez17|13 years ago|reply
[+] [-] jules|13 years ago|reply
https://en.wikipedia.org/wiki/Parkinsons_law_of_triviality
[+] [-] jdiez17|13 years ago|reply
[deleted]
[+] [-] viraptor|13 years ago|reply
[+] [-] gfunk911|13 years ago|reply
[+] [-] bayesianhorse|13 years ago|reply
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.
[+] [-] quacker|13 years ago|reply
[+] [-] mrtksn|13 years ago|reply
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 :)
[+] [-] moioci|13 years ago|reply
[+] [-] Falkon1313|13 years ago|reply
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.
[+] [-] jamessb|13 years ago|reply
Some people list off-by-one errors as the third hardest thing.
[+] [-] auctiontheory|13 years ago|reply
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.
[+] [-] tokenadult|13 years ago|reply
http://www.ams.org/notices/200502/fea-kenschaft.pdf
reports on her work in teacher training programs for in-service teachers in New Jersey. "The understanding of the area of a rectangle and its relationship to multiplication underlies an understanding not only of the multiplication algorithm but also of the commutative law of multiplication, the distributive law, and the many more complicated area formulas. Yet in my first visit in 1986 to a K-6 elementary school, I discovered that not a single teacher knew how to find the area of a rectangle.
"In those innocent days, I thought that the teachers might be interested in the geometric interpretation of (x + y)^2. I drew a square with (x + y) on a side and showed the squares of size x^2 and y^2. Then I pointed to one of the remaining rectangles. 'What is the area of a rectangle that is x high and y wide?' I asked.
. . . .
"The teachers were very friendly people, and they know how frustrating it can be when no student answers a question. 'x plus y?' said two in the front simultaneously.
"'What?!!!' I said, horrified."
Professor Kenschaft's article includes other examples of the mathematical understanding of elementary schoolteachers in New Jersey. In this regard, New Jersey may actually set a higher standard than most states of the United States, so all over the United States, there is risk of learners being misled into incorrect mathematical conceptions by their schoolteachers.
The problem is not ideally written, to be sure. In February 2012, Annie Keeghan wrote a blog post, "Afraid of Your Child's Math Textbook? You Should Be,"
http://open.salon.com/blog/annie_keeghan/2012/02/17/afraid_o...
in which she described the current process publishers follow in the United States to produce new mathematics textbook. Low bids for writing, rushed deadlines, and no one with a strong mathematical background reviewing the books results in school textbooks that are not useful for learning mathematics.
But if you put a poorly written textbook into the hand of a poorly prepared teacher, you get bad results like that shown in the submission here. Those bad results go on for years. Poor teaching of fraction arithmetic in elementary schools has been a pet issue of mathematics education reformers in the United States for a long time. Professor Hung-hsi Wu of the University of California Berkeley has been writing about this issue for more than a decade.
http://math.berkeley.edu/~wu/
In one of Professor Wu's recent lectures,
http://math.berkeley.edu/~wu/Lisbon2010_4.pdf
he points out a problem of fraction addition from the federal National Assessment of Educational Progress (NAEP) survey project. On page 39 of his presentation handout (numbered in the .PDF of his lecture notes as page 38), he shows the fraction addition problem
12/13 + 7/8
for which eighth grade students were not even required to give a numerically exact answer, but only an estimate of the correct answer to the nearest natural number from five answer choices, which were
(a) 1
(b) 19
(c) 21
(d) I don't know
(e) 2
The statistics from the federal test revealed that for their best estimate of the sum of 12/13 + 7/8,
7 percent of eighth-graders chose answer choice a, that is 1;
28 percent of eighth-graders chose answer choice b, that is 19;
27 percent of eighth-graders chose answer choice c, that is 21;
14 percent of eighth-graders chose answer choice d, that is "I don't know";
while
24 percent of eighth-graders chose answer choice e, that is 2 (the best estimate of the sum).
I told Richard Rusczyk of the Art of Problem Solving about Professor Wu's document by email, and he later commented to me that Professor Wu "buried the lead" (underemphasized the most interesting point) in his lecture by not starting out the lecture with that shocking fact. Rusczyk commented that that basically means roughly three-fourths of American young people have no chance of success in a science or technology career with that weak an understanding of fraction arithmetic.
The way this is dealt with in other countries is to have specialist teachers of mathematics in elementary schools. Even with less formal higher education than United States teachers,
http://stuff.mit.edu:8001/afs/athena/course/6/6.969/OldFiles...
http://www.ams.org/notices/199908/rev-howe.pdf
teachers in some countries can teach better because they develop "profound understanding of fundamental mathematics" and discuss with one another how to aid development of correct student understanding. The textbooks are also much better in some countries,
http://www.de.ufpe.br/~toom/travel/sweden05/WP-SWEDEN-NEW.pd...
and the United States ought to do more to bring the best available textbooks (which in many cases are LESS expensive than current best-selling textbooks) into many more classrooms.
[+] [-] sillysaurus|13 years ago|reply
I.e. teaching students the steps to solve a math problem is not teaching them how to think about the problem.
I instantly knew 12/13 + 7/8 was ~2 because I visualize two pie charts in my head, both of which are mostly full. This is in contrast to the other way to solve the problem, converting the fractions to a common denominator and then dividing by the denominator. It would take me some time to do the latter, whereas I can instantly do the former.
I don't think the students who got that wrong (nor some who got it right) do any kind of visualization in their heads.
Teachers need to realize that it's the operations in the head that count the most, not rote memorization of steps to solve a problem.
[+] [-] DanBC|13 years ago|reply
These are released.
People can make corrections.
For something like math this could have significant impact not just in the US and EU but in the developing world too.
PS: About the fraction multiple choice: There's probably a bad joke about 24% being what we'd expect if we let the students chose at random. I'm not funny enough to think what it is. (The punchline being that there are 5 options, not 4.)
[+] [-] dhimes|13 years ago|reply
One doesn't seem to preclude the other, nor does it seem to mean you won't have success in a science or technology career. I think you'll find a lot of people who know how to solve, say, 'circular motion problems,' but don't really understand what they are doing.
[+] [-] jimmahoney|13 years ago|reply
[+] [-] blei|13 years ago|reply
[+] [-] GuiA|13 years ago|reply
[+] [-] utopkara|13 years ago|reply
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.
[+] [-] DanBC|13 years ago|reply
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.
[+] [-] dfc|13 years ago|reply
[+] [-] pbreit|13 years ago|reply
[+] [-] alexvr|13 years ago|reply
[+] [-] sosborn|13 years ago|reply
Kidding aside, this is probably a good demonstration of how shoe stringing our education budgets might not be the best idea.
[+] [-] king_jester|13 years ago|reply
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.
[+] [-] jostmey|13 years ago|reply
[+] [-] gbaygon|13 years ago|reply
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.
[+] [-] henrik_w|13 years ago|reply
[+] [-] unknown|13 years ago|reply
[deleted]
[+] [-] alan_cx|13 years ago|reply
As for the teacher, well, I and my entire class once spent half a lesson arguing with our maths teacher who was swearing blind that 1x1=2. She wasn't an idiot or any thing, actually usually a very good teacher, but she just had one of those silly mind blocks. Once it clicked in her head she basically realised how mad she looked and took it with great humour. So, fair enough. Only human.
[+] [-] noonespecial|13 years ago|reply
Add it all up and it only takes third grade math to know it equals fail.
[+] [-] Beltiras|13 years ago|reply
[+] [-] danso|13 years ago|reply
That said, this gave me a little glimmer of hope about the state of logic education, at least among our third grade students.