How Old Is The Shepherd?(robertkaplinsky.com)
robertkaplinsky.com
How Old Is The Shepherd?
http://robertkaplinsky.com/how-old-is-the-shepherd/
24 comments
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.
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.
>Clearly the problems she gets in school are extremely rote
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'.
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 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.
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.
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.
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...
[1] http://reason.com/archives/2013/11/22/no-arne-duncan-white-s...
I would be very surprised if that is not a rubric that was provided by an adult (whether teacher or otherwise.) Certainly, I saw this type of thing more than once when I was in school, and this was before the predominance of frequent high-stakes standardized testing on which school funding decisions were based, which increase the incentive for schools to narrowly prepare students to do well on the kinds of problems likely to appear on the tests.
Well look, all math problems have a numerical answer. I've been doing them for 8 years so far and every time they've had an answer. These problems are also very silly. But I can confidently state that every math problem has some words and numbers and you are expected to perform some kind of calculation with the numbers based on the words around them.
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.
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'm not going to pretend to be the most intelligent person here or even that I'm more intelligent than all of those kids who came up with a number. But what I've always been willing to do is question people. Like most kids, I was given plenty of word problems in school. And there were several times where I called out the teacher or the textbook author for either leaving out information or relying on assumptions that I couldn't comfortably make. In fact, now that I think about it, later in school there were multiple choice answers that went something like "not enough information to answer".
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.
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.
I know there are at least some curricula that have word problems with at least 1 bit of extraneous information to try and train that. It can be somewhat amusing to see kids try and fit the extra number into their calculations.
> I think there is an unfortunate temptation to look at the students and think "how stupid are they?"
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.
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.
Yeah, I wasn't calling anyone out. Or if I was, I was calling out my first reaction.
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?
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?
I think perhaps a better setup to the problem for a child might be to make a question that is possible to answer, but has extraneous information. Something like:
"There are 12 children, 3 bats, 15 balls, and 2 swings. If everyone is holding a ball, how many balls are left over?"
Filtering out extraneous information is an important and related skill, but its slightly different thing than identifying when the problem as phrased does not have the information needed to support an answer.
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.
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.
That's a great point. I suppose the problem that I see here is that schools often present problems as always having a solution, even when there's none to be found.
+1 ... between your comment and the one you responded to, both the "non-math" skills needed to solve problems in the real world are described.
I agree wholeheartedly that students need to spend more time dealing with the setup of a problem (thinking about how best to express a vague idea mathematically) and in dealing with problems that don't have any known answer.
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.
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.
Honestly, I think that what you are discussing is the most basic of analytical thinking, and transcends mathematics.
The students aren't stupid, just badly trained.
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?"
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.
This is the equivalent of teaching a programmer to "shut up and code", even though they may have objections to the proposed solution.
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.
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.
Isn't that the whole point? We are teaching children badly by always giving them clear-cut problems that always have an answer, when the world is full of fuzzy problems that may not have an answer.
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".
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".
everything has answer we just don't know what it is :-)
For a given problem, can you write a program to determine whether or not it has an answer?
One of the most important things I learned in college was to say "I don't know". Many students (even college students), will dance around a question trying to get a hint of the answer. I have no problems saying that I don't know, followed by asking the teacher what was the answer/solution.
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.
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.
While I think this is true, I see the same thing in adults in professional positions (when given information and asked to respond to a query based on it, they will recombine the information and use it propose an answer even if where there is no sensible relationship between the question and the information rather than seeking additional information or identifying that the information offered is not sufficient to support a response to the question given) disturbingly often.
The "Education" system doesn't teach sheeple to be critical thinkers. It trains a subservient citizenry. You don't want your slaves to be critical thinkers, just trained enough. The student is disciplined and trained to guess based on incomplete information given to them. We have got to teach children to build upon first principles.
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.
https://mises.org/daily/2937/
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.
https://mises.org/daily/2937/
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.
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.
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.
I think students (who are themselves a product of their parents and their culture) are also part of the problem. When students don't try, they fall behind, and when they fall behind, they have no choice but to make silly guesses.
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.
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.
> This is totally unsurprising. Most word problems in textbooks are poorly written and have little to no basis in reality.
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.
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.
"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.
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.
> 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.
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.
Yes, there are many poorly written problems, but on the other hand, even well-written problems can have little basis in reality. Spherical cows in vacuum, and all that. When a question asks "You throw a ball at 80 km/h at 45 degrees, how far does it travel?" you could argue endlessly about air friction, the exact height of your hand when you let the ball go, Coriolis' force, curvature of the earth, and the fact that you can never throw a ball that fast... but you are not being critical. You're just being obtuse.
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.
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.
>* ... but you are not being critical. You're just being obtuse.* //
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.
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.
> I fear you're making the calculation the end in itself (we have computers for that!)
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."
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."
You can grok most sorting methods remarkably well, though, by lining up a bunch of office junk on a big table and working through the sorting algorithm physically.
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.
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.
I just asked my 2 yr old this same question. He just responded with "yellow!" /facepalm
That answer isn't really any more wrong than "His age is 25 sheep per dog".
Ask a stupid question, get a stupid answer!
Like the AT&T commercial guy would say, "That's adorable."
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.
EDIT: Just watched the video. Scary.
This would never happen if there was a laptop in every classroom.
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.
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.
Part of the problem with this question is that many students today don't know what a "shepherd" is. Perhaps this is also a failure of education, but it isn't a failure of math education. It would be better to say a "farmer" owns these animals, since most children would have heard the word "farmer" at some point in their lives. Or maybe it would be better to say "your neighbor" has so many cats and so many dogs but then the kids would probably just guess some advanced age because crazy cat ladies.
> Part of the problem with this question is that many students today don't know what a "shepherd" is.
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.
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.
What a shepherd is is completely irrelevant to the problem.
...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.
...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.
Really? I don't see that as being part of the problem. I have no idea whether it's true or not, but suspect it depends largely on socioeconomic status and geography.
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.
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.
Wait, how would having a laptop in the classroom prevent this situation?
Well, they could google "How old is the shepherd" and find a link to articles discussing that there's no answer...
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.
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.
Well, if students could use the laptop during the test, since its a no-answer question that comes directly from well-publicized earlier research, at least some of them would google it and probably get the correct answer.
This doesn't, of course, address the fundamental issue that the test reveals, but it would impact the results of the test.
This doesn't, of course, address the fundamental issue that the test reveals, but it would impact the results of the test.
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.
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.
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.
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.
I also think that because they don't expect a teacher to lie they tell themselves there has to be a answer.
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.
---
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.
Everyone is clearly reading too much into the question. There is no shepherd. Our fictional herd here is actually an allegory for the struggle between the lower class "the sheep" and the upper class "the dogs". A sheep can only do as a sheep does, and despite its best efforts can never ever be a dog. The belief that there is a shepherd, there to control the dogs, is a false hope given to the sheep by the dogs themselves.
Besides, even if there was a shepherd, what do you think happens to the sheep anyway? That's right.
Besides, even if there was a shepherd, what do you think happens to the sheep anyway? That's right.
Nice.
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.
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.
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.
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.
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.
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.
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.
We don't need people who fabricate data and make unfounded guesses.
[deleted]
Actually the correct answer is the shepherd was previously an investment banker who is now going through a mid-life crisis. He's only been a shepherd for 3 weeks and he bought the entire flock as is.
He's 52.
He's 52.
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.
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.
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.
> After all, this is a fairly basic Fermi problem.
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...
[1] https://en.wikipedia.org/wiki/Fermi_paradox
[2] https://en.wikipedia.org/wiki/Fermi_problem
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...
[1] https://en.wikipedia.org/wiki/Fermi_paradox
[2] https://en.wikipedia.org/wiki/Fermi_problem
> 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.
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.
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.
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.
> how to think and process. The idiots that run this country do not want that
"No one is going to give you the education you need to overthrow them." -- Assata Shakur
"No one is going to give you the education you need to overthrow them." -- Assata Shakur
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?
Are you sure they really knew much more about a practical system of education than we do now?
> 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.
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.
"Never attribute to malice that which is adequately explained by stupidity."
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.
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.)
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.)
Or, conversely, the shepherd is 16 and they are the father's sheep. Nobody said they were his sheep.
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.)
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.)
A certain amount of inbreeding of livestock is OK, but your hypothetical shepherd is asking for problems. b^)
[deleted]
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.
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.
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!
EDIT: Too harsh! Tone down!
I wonder if I do an abstract version of this in code sometimes when I'm being lazy...
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.
It's important not to confuse understanding a topic with having been taught the topic. If someone understands dimensional analysis, sure, they won't make the mistake. But it's easy to imagine students who were taught dimensional analysis making these kinds of mistakes.
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.
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.
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.
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.
> 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.
I am not suggesting that we admonish students to think mathematically. I'm suggesting that we teach them to.
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.
Link appears to be down :(
I weep for humanity.
Well, looks like the kid on 2:22 is on to something.
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.