How Old Is The Shepherd?
robertkaplinsky.com
robertkaplinsky.com
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.
I don't see any evidence of that in the comments on the original article or here. All of it seems to be focussed on the process of education and the habits it forms and the psychology of teacher/student (or more general authority figure) relations.
But the point that I still think is an issue is that we don't know how much of the effect is about what the students _understand_ as much as it's about how they see the situation they are in. What incentives are there, and what is the student actually thinking?
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.
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.
[1] http://reason.com/archives/2013/11/22/no-arne-duncan-white-s...
Not necessarily. It could just be that she is just learning how to answer tricky questions by using logic to determine what's not exactly spelled out.
That's about the same age that I started getting word problems that don't tell you specifically what you need to do so you have to 'find the problem'.
I think this is a huge part of "mathematical thinking skills" that we claim we want to instill in students. It's a sad thing that most people don't really have a good definition of what a "mathematical thinking skill" is in the first place. Figuring out how to adequately and precisely pose questions is one of the MOST mathematical skills, but it rarely shows up in discussions of standards and curricula.
While it may be the case that the education system fails to do enough at both of these, I suspect it does better at getting students to identify extraneous information and use only what is necessary than it does to get them to identify and confidently report when a question posed has inadequate information to support an answer.
The problem I see right now doesn't have an immediate connection between the statement and the question parts. I have the numbers 125 and 5 and the question "how old is the shepherd?". I can see that the answer must be a person's age. The numbers refer to how many sheep and how many dogs are in the flock, which isn't really related to the shepherd's age. Still, there's probably some lateral thinking involved, or there's an omission, because I must obviously answer with a number.
If I add, subtract or multiply the two numbers in any order, I do not get a number that is plausible as a person's age. However, if I divide them, I get 25, which is totally a possible answer.
25 is also the number of sheep per dog. Assuming that that's a normal ratio for a flock, we can say that the shepherd's flock contains 5 flock-units. So if the shepherd started at 17 years old (very plausible in a rural setting) and was given a new flock-unit every two years (obviously as you get older you get to take on more responsibility), at age 25 the amount of responsibility would have grown to 5 flock-units.
This answer has a nice symmetry to it. Math answers have some sort of nice symmetry in them, so this sounds plausible. With the information given I can do no better.
Answer: 25 years old.
I do not think a reasonable justification is "Well all the other problems I had that looked like this had an easy answer so this one should too". And the fact, that several students were unable to kludge an answer together supports that.
>With the information given I can do no better. You can do a lot better! You can say that the question is unanswerable.
This is the equivalent of teaching a programmer to "shut up and code", even though they may have objections to the proposed solution.
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.
Teachers are much more assertive when you admit that you don't know, rather than being trying to prove your worth everytime.
Of course, this kind of maturity is hard for little kids, specially when the "differences in power" between the kid and the teacher are the biggest.
Given the environment, why wouldn't the children guess? For years, we give them homework and tests where a guess is always equal to or better than no answer at all. But that's nearly the opposite of how the world works, so we're clearly teaching them the wrong things. I don't see how revealing that is not "fair".
Poor kids, they are so afraid of being wrong! They would rather guess then take the time to figure out the fundamentals. They want to tell the teacher what he/she wants to hear regardless if it is a correct solution or not. I suppose that's what humans learn in a prison.
School Sucks. Enough with Civic Religion.
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.
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.
There's a difference between a basis in reality and a internal completeness.
I think the problem isn't the way "most word problems in textbooks" are written, except insofar as most are, in fact, internally complete and so students are not taught to verify that the question is complete and are instead encouraged to just throw math at the numbers in the problem until they get something that looks like an answer.
Pedagogically, there may be some sense to this at certain levels, but eventually it becomes problematic if students don't have the both the skills and the understanding that it is desirable to first evaluate whether the question is complete.
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.
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.
The point of the question is not to indoctrinate students that air resistance is negligible. It's to let students practice their knowledge of Newtonian dynamics so that they can later proceed to more complex problems (hopefully with air resistance) if they want. Context matters.
So what about the original problem? I agree that the students' expectation that "there must be an answer" played a role, but I don't think the expectation is necessarily a bad thing. There's a reason that all the textbook problems (are supposed to) have nice answers: it actually makes it easier for the students to practice and internalize the concept they're learning.
I disagree. One is being critical.
If however you try to use that as an excuse not to perform the calculation then you're definitely being obstructive.
It is absolutely right for a student performing such calculations to realise there are issues with it not being realistic. They should understand that it's an approximation and understand that even in a perfect, flat, frictionless world it's still an approximation [relativistic mechanics are not being used] but that nonetheless it's a useful model the result produced by Newtonian mechanics is useful.
Indeed I'd say the realisation and ability to express the limitations mark out a student as capable of analytical thought.
>it actually makes it easier for the students to practice and internalize the concept they're learning. //
I fear you're making the calculation the end in itself (we have computers for that!) and not the subject of critical analytical thinking which IMO defines mathematics.
Well, I have to disagree. Of course it's rather silly to make the calculation the goal in itself, but one needs to have a decent "feeling" of what would happen in a given domain. In math (up to college freshmen) or physics, that involves a lot of number-crunching. Until you get familiar with it.
It's same as CS students implementing quicksort, merge sort, heapsort, etc., even though when they graduate they will all use libraries. You can't really grok quicksort by reading the textbook and say "Hmm, I see."
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.
EDIT: Just watched the video. Scary.
This doesn't, of course, address the fundamental issue that the test reveals, but it would impact the results of the test.
Yes, the superior abstract reasoning that the computer imparts would definitely solve this!
(I'm also fairly skeptical of this finding).
From the article:
The “How Old Is the Shepherd?” question was popularized by an essay written by Professor Katherine K. Merseth in 1993 and was based on research by Professor Kurt Reusser in a paper presented at the 1986 American Educational Research Association annual meeting. It is noteworthy that Professor Merseth wrote that “researchers report that three out of four schoolchildren will produce a numerical answer to this problem.” Twenty years later that statement still held true.
Even if the children aren't familiar with the term "shepherd", it still comes down to the fact that they do not have enough information to answer the question and still attempt to perform some sort of calculation with no reasoning of why they are doing it.
What a shepherd is is completely irrelevant [1] to the problem.
> Perhaps this is also a failure of education, but it isn't a failure of math education.
The inability to identify which things in a word problem are relevant to the question being asked is clearly a failure in education in the area of logic and reasoning, if not specifically in "math" per se.
[1] Well, unless "shepherd" is some thing that has ownership of a number of dogs and/or sheep that has a fixed mathematical relationship to its age, but I doubt any child's lack of experience with the term would leave them with a prior expectation that that was the case.
...says the person privileged to know what a shepherd is. I agree that math education could be improved, but this study may not be the best basis for that improvement.
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.
I also think that because they don't expect a teacher to lie they tell themselves there has to be a answer.
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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.
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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.
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.
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.
We don't need people who fabricate data and make unfounded guesses.
I was greatly concerned for a second there, before realising I was confusing Fermi Paradox[1] with Fermi Problem[2].
Unless there's something the sheep are keeping very well hidden...
He's 52.
No, that would've been a great outcome in comparison. Instead, they attempted to answer it using only the information provided, showing a complete lack of reasoning skills.
Besides, even if there was a shepherd, what do you think happens to the sheep anyway? That's right.
I disagree though, the sheep invented the shepherd to give themselves a purpose.
History shows that a strong dictator can keep a population down for an awfully long time with nothing but violence. There's no reason for the dogs to bother inventing anything.
That point remains wrong. You can treat the question as a Fermi problem, but when you do that, it's because you realize and acknowledge that there isn't enough information in the problem to complete it by other means.
"No one is going to give you the education you need to overthrow them." -- Assata Shakur
Are you sure they really knew much more about a practical system of education than we do now?
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.
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.)
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.)
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.
EDIT: Too harsh! Tone down!
I am not suggesting that we admonish students to think mathematically. I'm suggesting that we teach them to.
I think it's a mistake to argue that because students don't understand method X to solve a problem, they should be taught method Y. The problem is the lack of understanding, not the method.
My daughter smiled and tried to hand me her juice. Granted, she's 15 months old...