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A “simple” 3rd grade problem
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- jostmey 13y agoThe 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.
- gbaygon 13y agoThe 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.
- bbx 13y agoIt'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).
- jay_m 13y agoThat water lily problem is very neat, hadn't heard of it before.
- georgemcbay 13y agoI 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.
- alok-g 13y agoMy 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? :-)
- mrtksn 13y agothe 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. so, +1 to the student :)
- moioci 13y agoWho was it that said the biggest problem in programming is concurrency and off by one errors?
- Falkon1313 13y agoOr 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.
- jamessb 13y agoWell, Phil Karlton said that "There are only two hard things in Computer Science: cache invalidation and naming things". Some people list off-by-one errors as the third hardest thing.
- randlet 13y agoI think the joke is: "There are only two hard things in Computer Science; cache invalidation, naming things and off by one errors".
- gizmo686 13y agoWhats the joke? The hard things are: 0) cache invalidation 1) naming things 2) off by one errors Looks like he counted right to me. EDIT: fixed newlines
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- oneeyedpigeon 13y ago[ "cache invalidation", "naming things", "off by one errors"].length != 2
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- kaeluka 13y agoI 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."
- hmottestad 13y agoCan 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?
- redblacktree 13y agoIf 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.
- deleted 13y ago[deleted]
- SilasX 13y ago>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).
- yew 13y ago
- viraptor 13y agoOh 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).
- gfunk911 13y agoAs which point they decide he is qualified to play a glockenspiel.
- bayesianhorse 13y agoThe 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.
- quacker 13y agoAbsolutely. 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?"
- gizmo686 13y agoThe 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.
- rquantz 13y agoYes, the teacher was trying to teach fractions, the student was doing discrete math.
- a-nikolaev 13y agoThanks, 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.
- downandout 13y agoThe 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.
- shurcooL 13y agoYou 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).
- downandout 13y agoTrue, 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.
- DanBC 13y ago
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- alexvr 13y agoI'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.
- sosborn 13y agoYes, if it was a different question then the teacher might have been right. Kidding aside, this is probably a good demonstration of how shoe stringing our education budgets might not be the best idea.
- king_jester 13y ago> "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.
- auctiontheory 13y agoA 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%." We're doomed.
- s_kilk 13y agoWait... How on earth do they justify redefining 50% as 75% ? (or whatever the actual numbers are)
- hashmymustache 13y agosounded like teachers were scaling performance to meet standards instead of adjusting teaching quality
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- GuiA 13y agoIt's called grading on a curve, and it's (imo, unfortunately) very common in undergraduate courses in the US :)
- katbyte 13y agoit was used by many of my math & science classes here in canada, and i'm unsure why it is a problem? The prof would make the test very hard so the average was around 50-70 and then use a curve to get grades.
- colomon 13y agoIn 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...
- utopkara 13y agoThis 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.
- dfc 13y agoAnswer: 1, 2 and 2 cuts.
- dhimes 13y agoIf I cut myself shaving in two places with one motion of my razor, how many cuts have I made?
- Kequc 13y agoThe 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!
- dfc 13y agoIf 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.
- randlet 13y agoAnybody 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.
- drtse4 13y agoNot 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.
- alok-g 13y agoSome 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.
- hashmymustache 13y agoMore like if she works just as slow, geez, 10 min to cut a board.
- rquantz 13y agoShe's using a nail file.
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- driverdan 13y agoI 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.
- jules 13y agoIt's called bikeshedding. The more trivial the problem, the larger the population able to discuss it. https://en.wikipedia.org/wiki/Parkinsons_law_of_triviality https://en.wikipedia.org/wiki/Parkinsons_law_of_triviality
- jdiez17 13y agoIt'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.
- jdiez17 13y agoIt'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.
- pbreit 13y agoOnly on HN would you find people trying to make the case that the question is ambiguous. What is the matter with you people?
- henrik_w 13y agoJust asked my 10-year-old the question. He thought for 5 seconds and answered 20 minutes.
- danso 13y agoThis 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.
- Beltiras 13y agoI 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.
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- saejox 13y agoThere are no assumptions about the size of pieces. I can do it it 2 secs. Just pinch 2 splinters of the board.
- fcsss 13y agoThis is too obvious to be interesting.
- RivieraKid 13y agoPerhaps 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.
- rsl7 13y agovery 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.
- pfarrell 13y agoTeachers cannot be expected to be perfect. Their responsibility is to educate kids of a wide range of abilities. I celebrate the fact that we have the ability to discuss this in an open forum. If you want to see how good you are at writing test questions with unambiguous answers, I challenge you to write a full set of questions for a trivia night at your local bar/church/whatever. I wager you will be pleasantly humbled.
- tokenadult 13y agoI 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," http://www.ams.org/notices/200502/fea-kenschaft.pdf 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_of_your_childs_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/ http://math.berkeley.edu/~wu/ In one of Professor Wu's recent lectures, http://math.berkeley.edu/~wu/Lisbon2010_4.pdf 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/www/readings/ma-review.pdf http://stuff.mit.edu:8001/afs/athena/course/6/6.969/OldFiles... http://www.ams.org/notices/199908/rev-howe.pdf 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.pdf 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.
- nijk 13y agohttp://www.suntree.brevard.k12.fl.us/Students/MathSuperStars/MathSuperStars.html http://www.suntree.brevard.k12.fl.us/Students/MathSuperStars... That is a SuperStars worksheet! It is an enrichment problem aet for gifted kids. We had those decades ago. And we also had teachers who had a weaker understanding of arithmetic than their students. The more things change...
- alan_cx 13y agoFor 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.
- serginho 13y agoIt' just an interpretation of a language. If it was - to cut out 2 pieces from an infinity board - then a teacher is correct. If it was - to cut into 2 pieces a board to get nothing from a board in the end - then a student is correct.
- noonespecial 13y agoThis 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.