
A “simple” 3rd grade problem - adito
http://math.stackexchange.com/q/379927/44971
======
downandout
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.

~~~
Total_Meltdown
It is open for debate.

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> )

~~~
downandout
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.

~~~
alok-g
The question says "board" while the illustration shows more of a rod. I deduce
the question to be at least one of inconsistent or incomplete.

------
bbx
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).

~~~
jay_m
That water lily problem is very neat, hadn't heard of it before.

~~~
alok-g
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? :-)

------
randlet
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.

~~~
drtse4
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.

------
driverdan
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.

~~~
jdiez17
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.

------
viraptor
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).

~~~
gfunk911
As which point they decide he is qualified to play a glockenspiel.

------
bayesianhorse
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.

~~~
quacker
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?"

~~~
gizmo686
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.

------
mrtksn
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.

so, +1 to the student :)

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

~~~
jamessb
Well, 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
I think the joke is: "There are only two hard things in Computer Science;
cache invalidation, naming things and off by one errors".

~~~
gizmo686
Whats 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

~~~
oneeyedpigeon
[ "cache invalidation", "naming things", "off by one errors"].length != 2

------
auctiontheory
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%."

We're doomed.

~~~
s_kilk
Wait... How on earth do they justify redefining 50% as 75% ? (or whatever the
actual numbers are)

~~~
GuiA
It's called grading on a curve, and it's (imo, unfortunately) very common in
undergraduate courses in the US :)

~~~
katbyte
it 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.

~~~
GuiA
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.

------
tokenadult
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,"

<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...](http://open.salon.com/blog/annie_keeghan/2012/02/17/afraid_of_your_childs_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.

<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://stuff.mit.edu:8001/afs/athena/course/6/6.969/OldFiles/www/readings/ma-
review.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.pd...](http://www.de.ufpe.br/~toom/travel/sweden05/WP-SWEDEN-NEW.pdf)

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
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.

~~~
dos1
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 :)

~~~
SiVal
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.

~~~
Periodic
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.

~~~
waps
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.

~~~
sillysaurus
You may enjoy Feynman's excellent talk "The Relation of Mathematics and
Physics": <http://www.youtube.com/watch?v=kd0xTfdt6qw#t=1m05s>

------
utopkara
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.

~~~
dfc
Answer: 1, 2 and 2 cuts.

~~~
dhimes
If I cut myself shaving in two places with one motion of my razor, how many
cuts have I made?

~~~
Kequc
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!

~~~
dhimes
You would call that one cut?

~~~
Kequc
The joke is that some people in this thread are contesting the correct answer
to the question in the OP...

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

~~~
wissler
I wish this kind of problem were only limited to HN...

~~~
pbreit
You sorta wish or hope that the intelligence here is a tick higher.

------
alexvr
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.

~~~
sosborn
Yes, 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.

------
jostmey
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.

~~~
gbaygon
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.

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

------
alan_cx
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.

------
noonespecial
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.

------
Beltiras
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.

------
danso
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.

------
serginho
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.

------
kaeluka
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."

~~~
hmottestad
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?

~~~
SilasX
>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).

~~~
hmottestad
Like that:

    
    
      -------------
      |     |     |
      |     |     |
      |-----------|
      |           |
      |           |
      -------------

~~~
hmottestad
Or you could say 11 minutes and do this:

    
    
      -------------
      |/          |
      |           |
      |-----------|
      |           |
      |           |
      -------------
    

Cut into two equal pieces and then cut off the corner :)

------
RivieraKid
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.

~~~
rsl7
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.

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

~~~
rquantz
She's using a nail file.

------
pfarrell
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.

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

------
fcsss
This is too obvious to be interesting.

------
nijk
[http://www.suntree.brevard.k12.fl.us/Students/MathSuperStars...](http://www.suntree.brevard.k12.fl.us/Students/MathSuperStars/MathSuperStars.html)

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...

