A “simple” 3rd grade problem

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.<p>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.

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).<p>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.<p>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!<p>This is a <i>3rd grade</i> math test that even includes an illustration of how the cuts are made! Within that context, the answer is unambiguously 20 minutes.

I have to laugh at how much play this got at Stack Exchange and here. This <i>is</i> 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.

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.<p>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.

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.<p>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.<p>so, +1 to the student :)

Who was it that said the biggest problem in programming is concurrency and off by one errors?

A friend of mine teaches school in rural North Carolina - here's what she tells me.<p>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%."<p>We're doomed.

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,"<p><a href="http://www.ams.org/notices/200502/fea-kenschaft.pdf" rel="nofollow">http://www.ams.org/notices/200502/fea-kenschaft.pdf</a><p>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.<p>"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.<p>. . . .<p>"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.<p>"'What?!!!' I said, horrified."<p>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.<p>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,"<p><a href="http://open.salon.com/blog/annie_keeghan/2012/02/17/afraid_of_your_childs_math_textbook_you_should_be" rel="nofollow">http://open.salon.com/blog/annie_keeghan/2012/02/17/afraid_o...</a><p>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.<p>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.<p><a href="http://math.berkeley.edu/~wu/" rel="nofollow">http://math.berkeley.edu/~wu/</a><p>In one of Professor Wu's recent lectures,<p><a href="http://math.berkeley.edu/~wu/Lisbon2010_4.pdf" rel="nofollow">http://math.berkeley.edu/~wu/Lisbon2010_4.pdf</a><p>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<p>12/13 + 7/8<p>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<p>(a) 1<p>(b) 19<p>(c) 21<p>(d) I don't know<p>(e) 2<p>The statistics from the federal test revealed that for their best estimate of the sum of 12/13 + 7/8,<p>7 percent of eighth-graders chose answer choice a, that is 1;<p>28 percent of eighth-graders chose answer choice b, that is 19;<p>27 percent of eighth-graders chose answer choice c, that is 21;<p>14 percent of eighth-graders chose answer choice d, that is "I don't know";<p>while<p>24 percent of eighth-graders chose answer choice e, that is 2 (the best estimate of the sum).<p>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.<p>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,<p><a href="http://stuff.mit.edu:8001/afs/athena/course/6/6.969/OldFiles/www/readings/ma-review.pdf" rel="nofollow">http://stuff.mit.edu:8001/afs/athena/course/6/6.969/OldFiles...</a><p><a href="http://www.ams.org/notices/199908/rev-howe.pdf" rel="nofollow">http://www.ams.org/notices/199908/rev-howe.pdf</a><p>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,<p><a href="http://www.de.ufpe.br/~toom/travel/sweden05/WP-SWEDEN-NEW.pdf" rel="nofollow">http://www.de.ufpe.br/~toom/travel/sweden05/WP-SWEDEN-NEW.pd...</a><p>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.

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.<p>How many cuts do you need to make in order to split a board into 2? How about 3? How about 4?<p>In this case, the teacher has failed. But, everybody must have learned something out of this.

Only on HN would you find people trying to make the case that the question is ambiguous. What is the matter with you people?

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.

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.

Just asked my 10-year-old the question. He thought for 5 seconds and answered 20 minutes.

For me this question is more about careful reading that actual mathematics. A valuable lesson, IMHO.<p>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.<p>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.<p>That said, this gave me a little glimmer of hope about the state of logic education, at least among our third grade students.

It' 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.

I would've arrived at the teacher's solution, but the question allows different interpretations and both answers are correct assuming different interpretations.<p>The <i>correct</i> answer would be "I do not know, this problem is under-specified."

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.

More like if she works just as slow, geez, 10 min to cut a board.

Teachers 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.<p>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.

There are no assumptions about the size of pieces. I can do it it 2 secs. Just pinch 2 splinters of the board.

This is too obvious to be interesting.

<a href="http://www.suntree.brevard.k12.fl.us/Students/MathSuperStars/MathSuperStars.html" rel="nofollow">http://www.suntree.brevard.k12.fl.us/Students/MathSuperStars...</a><p>That is a SuperStars worksheet!<p>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.<p>The more things change...