
How Old Is The Shepherd? - gwern
http://robertkaplinsky.com/how-old-is-the-shepherd/
======
hyperpape
I thint there is an unfortunate temptation to look at the students and think
"how stupid are they?", but I wonder if the issue is more about familiarity of
tasks.

If students are going along solving word problems that always make sense, they
have zero practice in identifying which questions are well posed and which
aren't. It may not be that they're bad at reasoning through word problems, so
much as they aren't used to word problems that make zero sense.

Given that context, it makes partial sense for a child to think "I must be
making a mistake--I can't see how this problem works, but it must work
somehow, because these problems always do make sense, even if sometimes I get
them wrong, because I am not smart enough to understand them."

Is that a good attitude? Heck no. If nothing else, this experiment shows that
we need to teach students that math problems aren't always going to be nicely
set up so that they make sense. But that's different from thinking kids are
dumb for not adapting to an unexpected wrinkle on a test.

An interesting companion experiment would be to prompt the students to explain
the problem first, or see whether they could accurately determine which
problems made sense and which didn't, when they knew that was what they were
doing. My prediction is that the results would not be great, but that they'd
be better.

~~~
MBCook
I think the last kid was _extremely_ revealing. There are two numbers, and
they didn't say "product", "sum", or "difference", so it must be division.

Clearly the problems she gets in school are extremely rote and don't really
require reading the problem much. Just look for a keyword or two and you can
'solve' it.

~~~
bentcorner
I feel bad for this child. Clearly she has gotten through her schooling
through sheer brute force without applying very much understanding. She is
likely at a grade level where building the basics back up is not possible
without 1:1 attention.

~~~
JshWright
Welcome to Common Core. Obviously there won't be non-sensical questions on the
test, so why bother exposing kids to them in class?

Teachers are being forced, now more than ever, to optimize for test scores, at
the expense of real learning.

This isn't a video about stupid kids, it's a video about a stupid education
system.

~~~
acheron
Opposition to Common Core? You must be a "white suburban mom"! [1]

[1] [http://reason.com/archives/2013/11/22/no-arne-duncan-
white-s...](http://reason.com/archives/2013/11/22/no-arne-duncan-white-
suburban-moms-arent)

------
NoPiece
Especially with younger kids, when they are asked a question by someone with
authority, there is a reasonable presumption that it can be answered. I'm all
for teaching them to challenge that presumption, but I think you'd get a more
fair result if the question was, "Can you figure out the age of the shepherd
if you know how many dogs and sheep he has in his flock?"

~~~
jcutrell
Agreed. I think this is also an issue with the way we teach; we should teach
students to have the confidence to deny the solvability or soundness of a
problem on their own (rather than forcing irrational thought in order to
arrive at a solution).

This is the equivalent of teaching a programmer to "shut up and code", even
though they may have objections to the proposed solution.

~~~
sailfast
Agreed - it would be interesting to run the numbers with "questions may not
have an answer" as a qualifier. If I try to put myself back in my young
student mindset, I would only expect to be correct with a non-answer response
for a specific kind of test, or with a certain type of teacher that asked non-
structured problems. If given this problem in a structured environment such as
a standardized test I would likely look for a possible answer, realize none
was possible, write a number as a guess with some notes about the median age
of shepherds, and then be extremely frustrated at the test designer haha.

I get that this is supposed to outline the differences between structured /
unstructured learning, thinking, and classroom conditioning, but it's not
quite fair to draw a conclusion that doesn't take that conditioning into
account.

------
x0054
The problem is teachers! One of my relatives is taking Community College level
math right now. I was helping her with some homework, which was multiple
choice. The problem was simple, find an area of a triangle. We did the
problem, and the answer was not among one of the possible solutions. I quickly
double checked my work, making sure I was right, and we decided to mark none
on the problem sheet and give the right answer + show work. The teacher said
that in this case you should always guess what the closest right answer is,
and just mark that. WTF. The actual answer was roughly 65, the closest number
to that on the answer sheet was 73.

Math in USA is taught like religion, you have to believe what you are told,
and god save you if you ask for proof. I remember in School we were reviewing
Pythagoras Theorem. I asked the teacher to explain why it worked. She
proceeded to draw on paper a triangle with sides of 3 and 4 inches, and then
measured the long side, and said ‘see,’ with a very proud tone in her voice.
When I asked if there is any more definitive proof, she replied that “it’s a
theorem, there is no proof, that’s just the way it is.”

I spent the first 6 grades of my education in Ukraine, admittedly in the
communist equivalent of prep schools. There we were given math problems, and
we were expected to solve them OR prove that they were unsolvable. At least
one problem on every test was unsolvable, and they actually expected you to
provide proof. My experience with college level math on US was the same, the
teachers were smart and taught well. But Middle School and High School level
math in USA is pathetic, in my experience.

~~~
dublinben
It sounds like you just had a bad math teacher. The Pythagorean Theorem has
quite a few proofs, a number of which are understandable to students with only
basic geometry knowledge. I'm almost certain I had to prove it in my math
class at the time.

------
epochwolf
This is totally unsurprising. Most word problems in textbooks are poorly
written and have little to no basis in reality. My experience as a grade
school student is that most teachers get annoyed when you question the logic
of the questions being asked. Critical thinking is not rewarded in those
situations.

So that given, which student is going to stick their neck out in that
situation? It's better to risk a wrong answer than to risk annoying the
teacher. The students may not think in those terms. They just know they need
to provide the correct answer in the correct format. Understanding why is not
important or rewarded. Only the result matters.

~~~
jumbled
"If two sides of a triangular garden are 3 and 4 feet respectively, find the
length of the third side."

Okay, hand me a tape measure.

In a middle school algebra class we had a word problem on a test that had us
divide by the number of cards in a deck of cards. I didn't think this was a
fair question, because it assumed that everyone knew how many cards were in a
deck of cards. I went to the teacher's desk and whispered this concern to her
and she said "Come on, even my 5 year old knows how many cards are in a deck
of cards."

Not 10 minutes later, another student raised his hand and asked how many cards
were in a deck of cards.

~~~
dragonwriter
> In a middle school algebra class we had a word problem on a test that had us
> divide by the number of cards in a deck of cards. I didn't think this was a
> fair question, because it assumed that everyone knew how many cards were in
> a deck of cards. I went to the teacher's desk and whispered this concern to
> her and she said "Come on, even my 5 year old knows how many cards are in a
> deck of cards."

Just for reasonably-common playing card decks, I can think of at least three
possibilities -- 48 (standard pinochle deck), 52 (poker deck w/o jokers), and
54 (poker deck w/jokers). And in middle school -- since there was a lot of
card playing in my house -- I probably would have been aware of all three.

------
jcutrell
This is incredibly interesting to me, and points to a problem in our schools.

We traditionally teach children to solve problems that have solutions. In
particular, eighth grade teaches pre-algebra and algebra, so the way students
approach problems is directly connected to how the student decodes the
language of a "word problem".

We need to change our conceptual understanding of what learning actually
requires, and give students a situation in which they must write the problem
themselves. Instead of "decoding" a predefined word problem, the student then
must explore the situation and interpret the available information.

A good way to handle this problem is to allow students to write word problems
for other students based on a set of data, and subsequently create the "answer
key" that includes the mathematical proofs. Proofs should be taught from a
much earlier age.

------
scragg
I just asked my 2 yr old this same question. He just responded with "yellow!"
/facepalm

~~~
mistercow
That answer isn't really any more wrong than "His age is 25 sheep per dog".

------
nimble
This would never happen if there was a laptop in every classroom. (EDIT:
Sarcasm alert -- see my response below before you downvote me)

EDIT: Just watched the video. Scary.

~~~
arbitrarilyHigh
Wait, how would having a laptop in the classroom prevent this situation?

~~~
nimble
That was sarcasm. My point was that if this level of critical thinking
deficiency is really present, then there is something seriously wrong with the
way kids are being taught, and calls that you hear for extra funding to pay
for things like laptops in the classroom are really a giant waste of money
until we fix whatever that is.

~~~
kbenson
That's a good example of Poe's law. I decided to reply with (hopefully more
apparent) sarcasm, because if it wasn't sarcasm, I didn't want to let it go,
and if it was, I didn't want to be an ass, and I assumed you could take it as
me going with the joke.

------
scotty79
I have an idea for whenever I'll have my own kids. I'll buy them as toys as
many measurement devices as I possibly can. This will show them, I hope, that
numbers are not an abstract thing. They are everywhere. Not visible at first
glance but with proper device you can see them.

I think the fact that my grandfather (electrician by trade and self-taught
carpenter, shoemaker and general DIY guy) owned and used vernier scale,
micrometer, folding rule, voltmeters and ammeters contributed to my ability at
mathematics, physics, chemistry, computer science and being a reasonable
person.

Surely I used a vernier scale as a makeshift futuristic gun, micrometer to
squish my fingers, folding rule as kind of pretend switchblade but later also
for seeing numbers in the world.

------
tedsanders
When I tutored sixth grade math, it was not uncommon to see kids who guessed
that 7 + 3 was 9. When you are that far behind in math, your only possible
approach to the subject can be weird heuristics that make no sense.

------
C1D
I don't honestly think that the students are "dumb", I think they tried to
work out the answer so they wouldn't fail the test with out trying. That's
usually my method, if I don't know how to work out the answer I'll guess or
try a logical method so that I might have a chance.

I also think that because they don't expect a teacher to lie they tell
themselves there has to be a answer.

------
rprospero
These kids were in something of a no win situation. I'm imagining a different
class where 75% of the students said that there wasn't enough information to
complete the problem:

\---

Our younger generation is doomed because they clearly don't understand math.
After all, this is a fairly basic Fermi problem. The shepard probably started
his own flock at around 18 and with two sheep. A sheep is a large mammal, so
I'll assume that it has a gestation period around 9 months. We'll also guess
that a sheep reaches sexual maturity after 3 years, or four gestation periods.
If we assume that all adult female sheep are kept gravid and that genders are
divided evenly, we know that the number of female sheep after n gestation
periods is

a(n) = a(n-1)+a(n-5)*0.5

Solving the recurrence relation and assuming that both sheep are adults when
we start, there will be 60 female sheep, or 120 sheep in total, after 19
gestation periods. Thus the shepherd is around 33.

Of course, I've made several assumptions in my solution and I wouldn't expect
eighth graders to come back with the same answer. However, the fact that they
just threw up their hands and declared that the problem is insolvable shows
that our school system is failing the next generation of entrepreneurs. We
don't need people who give up and declare that nothing can be done - we need
people who make guesses and take risks. Yes, expecting an eighth grader to
solve a recurrence relation is beyond their ability, but they could make
simpler approximations. Heck, they could have just divided the number of sheep
by two and been right to within an order of magnitude. It works as a model -
say a sheep is born every six month and be done with it. It may not be the
best model, but the perfect is the enemy of the good. Instead, we've taught
our student to be dependent on our teachers to provide them with all the
information and not make any guesses on their own.

\--

Obviously, the above is exaggerated, but the point remains that any answer or
non-answer by the students can be interpreted as a lack of math skills and
evidence of poor critical thinking.

I'm not going to deny for a moment that our schools do a horrific job of
teaching unit analysis - I've taught intro to physics and know how little our
high school graduates know. Still, students are caught between two masters.
When lacking information, students who guess are punished for lack of critical
thinking and those who don't guess are punished for lack of effort and
creativity. The blog post obviously indicates that the pendulum is currently
swinging toward rewarding creativity over critical thinking. However, the
pendulum will swing back and, in twenty years, we'll be complaining about how
are students aren't doing nearly as well as the brilliant, creative children
this blogger visited.

~~~
C1D
Is this honestly a joke. How on earth could you expect a student to come up
with an answer to a question with no information even slightly relevant to the
answer; that is not a basis to judge math skills.

Also your above calculation is no better than the ones the grade eights had
made. If you had any knowledge of farms you would know that the Shepard
probably bought a lot of sheep too and maybe some of them belonged to his
father. How can you assume that he started at 18. In Arabia some kids become
Shepards at the age of 13.

Your example was just plain stupid and so is your assumption that they have
bad math skills.

~~~
recuter
His assumptions may be faulty but that isn't the point, he constructed at
least a somewhat reasonable model and extrapolated some number crunching.

The OP link claims a few students tried (for no reason) to divide 125 by 5 and
failed to do even that. There is no right answer obviously, it is how you play
and tinker with it - the more mathematical tools you have in your toolbox the
more you could improvise, see also Credit Default Swaps.

Nerds like to conflate basic competence at technical skills with intelligence.
:)

However, while being able to muck about like this doesn't necessarily prove
one is smart it at least demonstrates some capacity for _learning_ and
exploration. I think that is the gist of the lament - that schools don't play
with math, they drill - and agree with its spirit.

~~~
chr1
His solution isn't much different from solutions most of the kids were giving.
That is when faced with problem you don't understand -throw away the problem
-solve something you are comfortable with -try to convince others it is the
solution of the original problem

We don't need people who fabricate data and make unfounded guesses.

------
lasermike026
We can learn from Descartes, Aristotle, and others to develop a curriculum
teaching children how to think and process. The idiots that run this country
do not want that. It's about the curriculum. We can develop the world we want
by teaching the next generation.

~~~
yongjik
I don't really understand this kind of categorical bashing of public
education. Descartes, Aristotle, and most "others" you're thinking of lived in
societies where the majority _couldn 't even read_, and they didn't consider
that a problem. Reading without moving your lips was considered an exceptional
skill during the middle ages (or so I've heard).

Are you sure they really knew much more about a practical system of education
than we do now?

~~~
niuzeta
> Reading without moving your lips was considered an exceptional skill during
> the middle ages (or so I've heard).

You are refering to what contemporaries wrote about Thomas Aquinas, a well-
renowned theologist in medieval era. I believe he was considered a _genius_
because he could read without moving your lips.

While it is a good food for thought for intellectual capabilities, one should
also take note that in his time books were not what books as we know. No
typography existed and whitespacing blocks between words were non-existent.
Truly, they were mostly transcribed words with sometimes bad spelling and
without a paragraph break. To read without reading out aloud meant that the
reader had a mental capability to quickly parse out words, form a sentence,
understand what the author meant out of context.

------
trinovantes
I remember back in my first year engineering physics midterm, it was 13
multiple choice questions and each question had a "there is not enough
information given" option. I guess that's why everyone at my school calls it
the rite of passage to university.

------
computerbob
Google Cache:
[http://webcache.googleusercontent.com/search?q=cache:http://...](http://webcache.googleusercontent.com/search?q=cache:http://robertkaplinsky.com/how-
old-is-the-shepherd/)

------
McLeopold
Let's assume the shepherd graduated high school at 18 then bought 2 lambs and
a farm. He was nice to his sheep so didn't breed them until 2 years and not
after 7. The average litter size is 2 and the average age is 11. Let's assume
an even number of male and female sheep. At year 2 he'll have 4 sheep (2
original and 2 new lambs). 2 more at year 3 and then 4 at year 3 (the first
litter can now breed). At year 7 the original sheep stop breeding.

The shepherd is about 30 years old after he has passed the 125 mark. The dogs
didn't matter. (Some story problems throw you a red herring.)

~~~
will_work4tears
Or, conversely, the shepherd is 16 and they are the father's sheep. Nobody
said they were his sheep.

~~~
McLeopold
You're right, nobody did say that. The point I was trying to make is that with
the absence of data, you have to make assumptions and they should be clearly
defined. My assumptions defined his starting age and how he obtained the first
2 sheep and the fact they were his.

I guess I would want my kids too look at any math problem with the following
things in mind: * identify units of measurement * identify useful data *
identify non-useful data * identify assumptions needed to complete answer

There are more or different assumptions I could make, such as cooperative
breeding amoungst other breeders or a larger flock to start with, or a
completely different model of breeding sheep. I could have described a
scenario where he started with 2 sheep and went through several trading and
auction rounds until he had 125 with some bonus dogs like the guy that got a
house from a paper clip.

We don't have a time measurement in years, so we need some type of sheep per
time or time per sheep unit. I choose an assumption of breeding and came up
with about 1 sheep per year per sheep.

The original question is very close to a nonsense question, which requires so
many assumptions that probably most answers can be well defended with the
right set of assumptions. But at the very least you would hope a student could
identify an answer in years could not be obtained with the given data of # of
sheep and # of dogs.

(And, no, I don't think the units method is beyond an eight grader, even if
the educational system thinks it is. The fermi problem may be.)

------
pfarrell
the 12th fibonacci number is 144.

Assume some of the sheep didn't make it through a winter or that one of the
dogs attacked some of them and had to be put down. So, 125 sheep left from a
potential of 144 after 12 years of shepherding seems possible.

Assuming the shepherd lives in a country that still has shepherding as a
profession, we'll assume s/he started at 15.

The shepherd is 27 years old

Q.E.F.

~~~
phaemon
What's scarier than the kids in that video are the number of people on HN who
don't seem to have grasped the concept of where "lamb" and "mutton" come from.
:-)

EDIT: Too harsh! Tone down!

------
hopp_check
I wonder if I do an abstract version of this in code sometimes when I'm being
lazy...

------
jessaustin
I think many eighth graders would be sophisticated enough for some dimensional
analysis if it were presented in the right way. It's hard to imagine students
who had been taught that, making these sorts of mistakes.

~~~
mistercow
But dimensional analysis is just another mathematical process turned into a
rote method which does not require deeper understanding. The problem is not
the pile of rote techniques is too small, but that students aren't learning
how to actually think mathematically.

~~~
jessaustin
I find this "think mathematically" concept you invoke to be a bit fuzzy. Even
geniuses (not me, but I've met some) can be caught out by novel constructions
if they haven't had time to consider them carefully. Much mathematical
sophistication consists of adding to "the pile of techniques". If a student
has too shallow a pile, the goal of educators should be to pile on more. If we
suspect the research under discussion represents something real and not simply
a series of methodology blunders, then it points to an improvement that
educators could make. It's likely that basic dimensional analysis is not a
complete answer, but I suspect that an admonition to "think mathematically" is
not even wrong.

~~~
mistercow
> I suspect that an admonition to "think mathematically" is not even wrong.

I am not suggesting that we _admonish_ students to think mathematically. I'm
suggesting that we _teach_ them to.

------
programmarchy
This is fascinating. I would be intrigued to see the results of asking the
same question to students undergoing self-directed learning, either in a home
schooling setting or in more progressive schools.

------
j2kun
Link appears to be down :(

------
RyanZAG
I weep for humanity.

------
gislifb
Well, looks like the kid on 2:22 is on to something.

------
benched
I would really, really like to see more HN readers asking their young
children, and posting the results.

~~~
JshWright
Your first data point:

My daughter smiled and tried to hand me her juice. Granted, she's 15 months
old...

