The student is absolutely correct. I don't think it's even open for debate. Cutting anything in half requires exactly one cut; cutting in thirds requires two. It's as simple as that. The teacher that crafted the question, or worse yet, the publisher of a textbook that may have provided the test question, needs to take a hard look at whether or not they are in the correct profession.
The fact that the teacher not only marked the answer wrong (which could have just resulted from looking at a publisher-provided answer key) but actually wrote down a completely incorrect justification for the teacher's incorrect answer is rather disturbing to me. Also, this did not occur in a vacuum. Either no other students answered the question correctly, or the teacher saw the question being answered correctly by others and repeatedly marked it wrong with the same justification. Either way, it causes concern about the teacher.
You can't tell if it was a simple "whoops, I thought this question belonged to a problem category X, and I overlooked that it does not" typo-like mistake, meaning the teacher would instantly realize his/her mistake if you point it. Or if they wouldn't get it even after you try to explain it to them (what you're trying to imply here).
When grading things, ppl usually face hundreds of copies at a time and it's very tedious. It's easy to scrutinize a single highlighted problem that someone got wrong in hindsight, not realizing the person might've only dedicated 7 seconds to this problem out of 1000 others that were graded correctly.
I personally try to give them the benefit of doubt and assume best case scenario (but I also understand it might not be).
I agree with this. I think it's a pretty decent example of the fundamental attribution error.
The context of the problem clearly sets it up as one of those "everyone gets this wrong, so make sure you think a second" situations (I had to think a second, anyway).
I'd extend that to the publisher as well, though (or at least it's individual employees creating the book). First off, assuming the answer key has "15", I'm not in any way saying it's okay that we have text books teaching clearly incorrect information; I'm also not in the know on how 3rd grade math textbooks are created. That said, I've been in tons of jobs where you're expected to produce a crap ton of work at a breakneck pace and god help you if you want five minutes to check your work for dumb errors, because y'aint gettin it.
Of course, it could also easily be said that this is also the publishers fault for creating a working environment that isn't sufficiently rigorous or overburdens the employees.
On the topic of how math textbooks are created, you might like this commentary from Richard Feynman when he served on a school-math-textbook recommendation committee:
"In algorithm classes, there are often many right answers"
I accept this, but I'm assuming the original algorithm that a creative student produces will pass tests/produce same output as the 'textbook' solution. My understanding of the original article is that a correct final answer was marked wrong.
"Same goes for most college math."
Absolutely. My favourite from 16+ maths (GCSE in UK) is the area of a trapezium. Most find the mean length of the two parallel sides and multiply that by the distance between the parallel sides (so make a rectangle of the same area). About 1 in 15 break the trapezium up into two triangles and add the areas.
The third option would be to chop off both "wings" and consider it as a square plus two triangles. That is often my instinct when I forget to do it with the mean length of the parallels.
True, but I think that maybe 1/3 or more of the students would have answered this correctly. At some point during the grading process, you would think the teacher would begin to wonder why all of these students would have answered such a seemingly simple question "incorrectly" and take a second look.
It's not a trick question, just a bad one. It doesn't include enough information to give an informed answer. Telepathy is required to suss out the author's intention.
In the real world, you can ask more questions and get a more complete picture. On an exam, generally you must accept what you are given.
I'd be less worried if this seemed like a one-off thing (or if math professors were obligated to drive exclusively on bridges designed by their own students, heh).
As it is, this is one case among many (not all about grade schoolers and not all 'stories on the Internet' by a long shot) and the professor doesn't always acknowledge they were wrong. Speaking as an engineer, the work is hard enough when you do understand the math.
My god, this looks more like a 4chan troll post than stackexchange. I'm not convinced this really happened. Is this the only kid in the class that got it right? Did the teacher not then notice when the brighter kids were coming up with 20 min that there may be something to it, and reconsider the question himself/herself? So much fail in so little space. Ugh.
In high school geometry, I remember my teacher making some assertion that was plainly false - I think it was that 3 planes always intersect in a line. After arguing with him for like 15 minutes, I walked to the front of the class and wrote a proof on the board. I spent the rest of the class period sitting outside.
You're not convinced this actually happened? When I was in elementary school, I regularly (ie. several times per semester) got into arguments with my math and science teachers over stuff this dumb. There's no need to make up something like this when you can find it in the real world so easily.
In grade school I had an argument with my science teacher about wheels. She said that a point along the outside of a wheel moved faster than a point nearer to the center (which is absolutely correct). However, she followed that up by saying that the outside of the wheel makes more revolutions than the inside. I tried to correct her, but she wasn't having any of it. So I grabbed my bike from outside, brought it into the classroom and tied two pieces of string onto one of the spokes on the bike: one near the axel, one near the tire. A few spins of the wheel had her convinced, but I can't believe I actually had to do it.
If anything, the inside should have to make more revolutions. Since the whole bike is traveling at 10 MPH, at the point with the smaller radius, you'd need more revolutions to travel that same linear speed.
A fence is made using 15 posts spaced equally along a straight line. There are 3m between each post. What is the distance between the first and last post?
When I'm teaching this kind of thing, we go out and walk around the building site opposite with a few 15m measuring tapes. The physical walking out and measuring helps.
Change "There are 3m between each post." to "The distance from one post centre to the next is 3m"
Which illustrates the general point: you need teams working the test and checking their answers against what the writer thought the answers were. You also need English specialists checking the wording of the questions.
The question does not say cut "into thirds," it says "into three pieces." This - http://i.stack.imgur.com/kEjP0.png - is a perfectly reasonable answer which, assuming the rate of cutting is constant, would result in 15 minutes.
It's a bad question.
Edit: That said, I would have given the same answer as the student, because I think that's the most reasonable interpretation, especially considering the illustration. But the keyword there is "interpretation." The question is ambiguous.
You are correct, but you can trisect that piece of wood an infinite number of ways. The logical equilibrium point is 3 even pieces.
The student chose a ratio of 1,1,1; which is the logical equilibrium point. your image shows 1.5,0.75,0.75; which is the second most logical ratio because it is in the form x + 2y = 3 (which can be trisected an infinite number of ways while maintaining that ratio). The third form would be x + y + z = 3; which can also be trisected an infinite number of ways and would be the least intuitive.
i am agreeing with you, i am just trying to show that it is illogical for it to be 'open for debate'.
There is a game theory term for this type of equilibrium, but i forgot its name. Its the same type of equilibrium as "there are three colors and a number, which one is different?" type sesame street problems.
I didn't see the picture at first, and reasoned just as you exposed.
However, the teacher corrects it by writing "4 = 20". This is plainly wrong and with no possible explanation, since following the above reasoning, cutting in 4 pieces would require: 10 + 10 / 2 + (10 / 2) / 2 = 17.5 minutes.
I can cut a piece of wood into thousands of pieces in 5 seconds. I just slice it across the top a few times with my saw, and all the sawdust that comes off counts as separate pieces.
As someone said below, it's only open for debate if you want to be pedantic. The Dr. Sheldon Cooper's among us may debate it, but it's pretty obvious what the question was looking for. There is even an illustration showing the cut, which would take an identical amount of time.
But even with your picture the answer can be 20 seconds. You're assuming the person is starting at the top of the line and cutting all the way through the board to the bottom. But they could just as easily rotate the board 90° and cut across a different axis. Assuming a 1" thick board, this means they're cutting through 1" of wood on each cut, meaning both cuts take the same amount of time.
this was my first thought, but then I realized the teacher gave justification for his answer.
the problem is poorly formulated. The teacher would have been correct if it had said "it took 10 minutes to cut away 2 pieces from a very large board (thus resulting in 2 cuts, 3 pieces total)", whereas the student's answer assumes a single cut, which is more reasonable.
Most likely, whoever produces and publishes the question sets changed the question for a new edition and did not notice that the new question also changed the answer.
In the previous edition it was probably something like "Marie works in a factory which makes cars; it takes her 10 minutes to finish two cars. How long will it take Marie to finish three cars?"
And the answer to that would be 15 minutes, and the reasoning in the answer (based on reducing fractions, which is what it's probably supposed to teach) would be correct.
But probably in the next edition the question changed from putting things together to cutting them apart, and the author/editor simply didn't realize that these are not interchangeable. The teacher, meanwhile, probably didn't look too closely at it, and simply applied the answer and reasoning supplied in the teaching materials for the question set.
None of which implies that the teacher can't do the math; rather, it implies systemic problems in the way the materials are produced and in the methods used by teachers to grade the work.
While, given the picture presented the child's solution makes the most sense, there is another close scenario in which the teacher is correct[1]. So its not really "as simple as that."
this is not a plausible scenario at all. if the quantity of work done is not equal at each step the question is impossible to answer which means that the teacher is nor right, just there is a possibility to be right. but then, every answer could be right, just adjust your cutting path according to the answer you would like to be correct.
Yes, the question did not account for unknown specifics. There is also a 3rd answer which in which the answer accounts for the person making the cuts having to answer the phone half way through the job.
Context is everything, this is a question for a 3rd grader not someone who is in Calculus. I think it's safe to assume the student is right given the screenshot.
It is ABSOLUTELY open for debate, and part of the clue is in the question "if she works just as fast" ie. the cutting rate is constant. Then, it is ambiguous since the SIZE of the pieces is not mentioned.
It's not the teacher's fault, per se; the question is unanswerable. The student picked one interpretation but the (likely) correct one is shown in the answer http://math.stackexchange.com/a/380007
It's only open for debate if you're being extremely pedantic. There is even an illustration demonstrating exactly what the cuts look like!
Yes, if you want to be very nit-picky the question is undefined, but this is a third grade math test and the only reasonable answer is 20min. The student was absolutely correct.
I agree. If I am the principal and the student protests to me and the teacher gives some bizarre-ass interpretation of the problem to me, the teacher will have a problem with me.
I've seen enough of that in high-school physics where the teacher literally doesn't understand what the hell the problem is asking to piss off the Good Humor man.
There is a picture of that illustrates the cut that is being made well enough to infer.
Some inferences have to be made, this is a human taking the test not a robot, and it's a 3rd grade test.
Even if the size of the pieces were specified, she could be cutting a different kind of wood or using a different saw or the humidity level could be different, but she's still "working just as fast".
3rd graders would be confused if you attempted to be completely unambiguous with this time of question.
Look at the image provided with the question. It shows a saw cutting the board in a way that leaves it unambiguous - the cutting time for that cut would be identical regardless of where each cut took place.
In reality, it probably was not open for debate. This is third grade math, remember. The student probably spent the last three problems working similar questions covering ratios/fractions. And the student probably spent at least a few weeks of school working similar types of problems on homework. After so long taking these types of classes and tests, if a student answers a question without using knowledge gained from the class and gets the question wrong, well...what did they expect?
(Not saying I like it, but it's the way it is many places.)
I would assume that the "*" next to the problem was an indication that this what not a 'time to fill a water bucket' problem and that more thinking would be required than the previous 3 problems.
If it is ambiguous, there is no answer. There must be an answer. Therefore, it cannot be ambiguous.
The answer given is the only one it is possible to give. Therefore, it must be the correct one.
The context isn't so much "third grade" as it is "math test", and very, very few math tests allow "Question ill-formed as posed" as a valid answer. Maybe more should.
That's actually a great idea. If I were a math teacher, I would teach my class that IFQ is a reasonable answer to a question, and I'd throw in a few plainly ill-formed questions just to keep them on their toes. Actual thinking > correct answers.
I think for most questions that are not very straight forward, IFQ would be a valid answer with only very little argumentation. That's why formal languages are needed.
The first round of the UK Maths Challenge * is multiple choice, and does often include questions with "not enough information provided" as one of the available answers. However, this isn't a mechanism for identifying badly phrased questions.
* (the feeder competition for the British Mathematics Olympiad, and then the International one)
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).
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? :-)
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.
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 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 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%."
>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.
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.
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...
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%).
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.
It's always interesting to hear how people go about solving math problems. You mention a pie chart visualization and then the much more labor intensive (but maybe "correct"?) method. I used a third way, which was thinking that 13/13 would be one, so 12/13 is pretty close, so that's ~1. And 8/8 would be 1, so 7/8 is pretty close and also ~1. 1 + 1 = 2 :)
I imagine there are myriad other ways people approach estimation problems like this. In response to the rest of your post, I was never taught how to "think" about math. I was educated in a decent school system, but it was all rote memorization of multiplication tables. I think most people who are interested in learning will come up with their own tricks regardless of curriculum. Of course, imagine how much better I'd be at this stuff if I had math teacher's who were competent :)
I use the heuristic that many of us here probably use, consciously or not, after our years of experience with math problems: if it's a math problem, as opposed to problem in some other domain that ends up requiring math (science, accounting, carpentry, etc.), there will be some degree of artifice in the problem. Somehow, the numbers will just happen to end up being integers or perfect squares or exact multiples or whatever, so that there is an easy way to solve this specific problem (not a general problem of this sort but this specific instance).
In this case, you examine the numbers and spot that they are both just "one off from one" fractions, so the sum is roughly 1+1. The test givers will then see to it that there is only one answer that matches the result of the "trick" they were testing to see if you could find.
Kids who get a lot of math internalize this heuristic, which actually trips them up briefly when they start having real science classes, because they think they've done something wrong if the answer turns out to be 5.6293 or 0.07291 instead of 4 or 9 or 5/8 or sqrt(10). They assume they missed the trick.
I've been tutoring 1 on 1 middle school kids (at the 8th grade level) in underprivileged areas and what I find is that they have little understanding for what fractions are/represent at all. For example, I asked a student what the decimal value of 1/2 was (I had explained what decimal values were beforehand) and she didnt know. I was shocked (maybe its because we're cs people we have a special affinity for 2s). As a further test I gave her a piece of paper and asked fold the paper in half and she knew it instantly. I then asked where on the paper the 1/4 mark was. Again, she didnt know. This came from a further problem of not understanding that (1/2)/2 is 1/4. After playing with the paper folding it in so many ways she started to internalize what these fractions meant.
When I did my undergraduate degree in physics I think one of the best things I learned early on was estimation skills. I was used to doing things precisely and finding the tricks. Our professors made jokes about things just needing to be right to "within an order of magnitude", and it wasn't for two years that I internalized that.
When you deal with the real world there are always a lot of errors and uncertainty in measurement. Simply being within 10% of the right answer is generally sufficient and quickly getting that answer over getting the 99.99% accurate answer is better if it takes you one-tenth the time.
This is something I find tremendously useful in programming, but at the same time find a lot of other developers amazed when it's used.
I don't care if the dataset in memory is 553MB or 632MB - what I really need to know is whether it's "a few tens of MB", "a few hundreds of MB", or a "a few thousand MB".
I don't care if the API server can service 7321 simultaneous requests or 6578 - I just need to know if its "a few hundred", "a few thousand", or "a few tens of thousands".
You can solve an enormous number of engineering and architecture problems with a reliable order-of-magnitude estimate - at the very least you can quickly exclude solutions that are vastly under (or over) provisioned for the problem you're trying to solve.
A good order-of-magnitude estimate is also a great error check for a more detailed calculation, if my quick estimate said "5000-ish plus or minus 50%", and your calculation says "24,152", one of us has got something wrong.
I remember this from my first university physics class. We would derive a movement equation for a cannonball, to find the optimal angle to shoot a cannon for maximum travel. Everybody knew the answer of course, but we'd always just used the formula. This time we'd start with the obvious integration equation, movement + attraction between 2 point masses, integrate over flight time, and find the point where it crosses the ground plane.
And then the teacher just took the range from the integration, and the formula, multiplied the two and put a ~= sign between them. I believe I actually stood up and said you can't do that and we had the first of many discussions about exactness.
That was scary.
That was my first run-in with what I considered the central article of my then faith : that you can derive the structure of the physical world from first principles. Throwing away terms in an equation in order to arrive at correct physics laws, I don't know, I considered it sacrilege or something. Of course I've since learned that deriving all of physics from it's own basic laws doesn't work, and the way we fix that is that we delete "inconvenient" terms in the equations when required. Deriving physics from a few mathematical laws is completely impossible. You can't even correctly derive the (mathematical) fields used in physics, so the very numbers that one uses to do physics aren't actually valid mathematical numbers.
So the relation between physics and mathematics is not that one is based on the other, because that was tried and didn't work out, and people have almost completely given up. So it was replaced by a marriage of convenience (this works ! Sure it won't validate mathematically but the numbers look really similar), ignoring at least a dozen elephants that stood in the way, and we just act like they don't exist.
As I read the question, I was prepared to work it out the hard way, and when I got to the part about not choosing the exact answer but only the closest, I was really dismayed because I thought I'd also have to work out how close each choice was to the actual value. Then I saw the possible answers and I thought, "Oh, 2."
I did it that way too. And while I agree that the best teacher for a person is that person themselves, this is such a basic understanding of math that I think that even the less motivated need to know this.
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.)
The joke would be that if you don't know the answer, the "I don't know" answer is completely correct.
More seriously, yes, I think the open source books (actually teaching materials that include books) will eventually replace commercial materials in almost all cases except those tertiary (college/uni) level classes where the book is written by the teacher. Financial pressure, if nothing else, will have this effect. Many of the open source books could be primarily the work of a single Benevolent Dictator For Life, of course.
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.
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.
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!
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?
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.
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?
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.
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.
The correct answer would be "I do not know, this problem is under-specified."
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
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!
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.
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.
>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.
> 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.
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.
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.
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.
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 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.