How do you think 32 eighth grade students would respond to this nonsensical question: “There are 125 sheep and 5 dogs in a flock. How old is the shepherd?”* Take a guess as to what percentage of them would give a numerical answer and then watch the video below:
Of the 32 students I interviewed, 75% of them gave me numerical responses. Going into this, I predicted it would be closer to 50%. Here are some of my observations from the 32 students:
- 2 students calculated the answer to be 130 (125 + 5)
- 2 students calculated the answer to be 120 (125 – 5)
- 12 students calculated the answer to be 25 (125 ÷ 5)
- 0 students calculated the answer to be 625 (125 x 5)
- 4 students stated that they guessed their answer (90, 5, 42, and 50)
- 4 students tried to divide 125 by 5 but could not correctly implement the procedure
Three particularly interesting students included:
- The student who found the shepherd’s age by using the old “add the sheep and dogs and divide by two” trick and got 65. You didn’t know about that trick?
- The student who got 120, then said that you don’t have enough information to figure out the shepherd’s age, then seemed to feel so uncomfortable with that conclusion that the student decided to guess 90 years old.
- The last student who explained that the reason for dividing was because it didn’t say “sum,” “difference,” or “product” in the problem.
Something I had not thought of until Cari Noble mentioned it in the comments was the idea of intellectual autonomy. Many of the students knew I was crazy for asking them this question, yet they felt like they couldn’t question the questioner and were obligated to answer me. So, they repressed that feeling and gave me an answer anyway. That’s certainly not where we want students to be. I do understand that it is important to be respectful of authority figures, but you can be respectful and still express your thinking.
As for Math Practice 1 from the Common Core’s Standards for Mathematical Practice, here are expanded versions of the quotes from the video:
- Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution.
- They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt.
- Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?”
There’s no better way to see if your students are making sense of problems than to ask your students this same question. You have to make sure you keep a straight face and have them write their answers on a piece of paper. When everyone is done, have them crumple the paper into a ball and throw it at someone else in the classroom. After a few rounds of this, the papers will be sufficiently mixed that you can then ask each person to open up the paper and read what was written to have a conversation about their classmates’ answers. When you’re done with the conversation, show them this video so that they realize they’re not alone. You can download it by clicking the “Download Video” button below. After you try it out with your students, please post a comment about how it went along with the students’ grade level.
*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.
Wish you’d warn me that this would be a depressing video. I’m now curious to see how my kids would do. I can’t believe the 75% and 100%!! That’s crazy. But the answer really isn’t 25?
Hahaha. I guess if this was a question for Who Want’s To Be A Millionaire, this would not be the one you want to use for polling the audience.
Are you sure the kid who gave you 42 as the answer was just guessing and not making a Hitchhikers Guide reference? Still, not making sense of the problem per se, but somewhat awesomer all around. 🙂
OK. That kid gets a pass then as it is the answer to the ultimate question of life, the universe, and everything. Actually though I do remember asking the student how he got it and he told me that it was “a long guess.” That reply caught me so off guard I forgot to have him
rightwrite it down.Write 🙂
Oops, yeah “write” it down.
“A long guess”? Yeah, that’s a hitch-hiker fan. That kid wins.
P.S: The correct answer is clearly 23-27 years old, assuming he started shepherding at 16-18 and roughly doubled his flock every year. But how many golf balls *can* you fit in a school bus?
The golf ball question is quite different from the shepherd’s age problem…almost everyone knows you can fit ‘all of the golf balls’ in the bus.
I know this is 75 because i have a much stronger reason. In the simple word to i explain. The question is about to what is the age. There is the age means experience not the real age. Because in a flock there is totally 130 animals sheeps and dogs but the thing is these animals not can controlled by 1 person. Normally required person is to be 5 to controll then it means 5 people controlling this. This people are all called shepherd. So each person age is include in question means by the combination of 5 persons experience make the age and the average age for controlling sheeps has to be 15 years old atleast this all persons experience make the age 75.
What if the big bad wolf got some of the sheep? Certainly one or two sheep per year would go “missing”. :p
I think that it takes a certain degree of courage and self-confidence to contradict a teacher. The student’s main objective is not to make sense of the question, or even to answer it correctly, but to give an answer that will satisfy the teacher. Students believe that an honest answer (“I don’t understand the question”) would not satisfy the teacher, so they make random guesses instead.
Great point Dave. If that is happening in classrooms, that is also a problem worth reflecting on. Avoiding contradicting the teacher should not take precedence over reasoning.
Then I’d suggest something with redundant data to filter the ability of thinking from the ability of contradicting the teacher, something like “If 10 days before concert 6 tickets cost 100$, how much cost 4 tickets if 3 people pays it 2 days before the concert?”
Building off of Dave’s remark, what do you think the responses would have been had you had them write them down or post them on a blog? Do you think that taking the face to face contact would take away the nervousness of contradicting a teacher? I think I will try it with students!
I really don’t know Maggie. If you try this out, I’d love it if you report back and share what you learned. I never tried that so it’s hard to be sure.
This is fantastic–I’m going to include it in a post I’m writing about CCSS, and I submitted it to Reddit so more people would come read it.
Perfectly illustrates the difference between being able to calculate correctly and being able to reason mathematically. Well done.
Thanks Hollis! You might also like this post then: http://robertkaplinsky.com/what-does-it-mean-to-understand-mathematics/
Wow, surprised, and not in a good way 🙁 that these are 8th graders messing up on this… couple of random thoughts: I wonder if it’s possible the word “shepherd” simply isn’t familiar to many of these kids (it only comes up in certain contexts these days)… the word sounds kinda like “sheep” so they may relate it to that, or it sounds like “German shepherd” a dog they are likely familiar with, and maybe that’s confusing them if they don’t know the real reference…? (not that that helps explain their numerical answers, just their confusion)!
Yes Shecky, some did ask about what shepherd meant. Some thought it meant “sheep.” Also some thought a flock was only for birds and didn’t understand how there were only sheep and dogs. Still, even if the question was asking about how old the sheep are, they shouldn’t know the age either.
I thought about changing the question to make it more relevant to urban students, but I wanted to keep the question the same as it had been in the research from about 25 years earlier to make it more comparable.
I emailed this link to the middle school teachers at our school and I am interested in their replies. One of the comments above suggested that the root of this is a student’s desire to simply satisfy their teacher in some way. This is a very real – and very depressing – possibility. I often hear this form of a conversation with regard to writing for different teachers. There is enough inconsistency in how writing is graded that students think they need to just figure out what a certain teacher ‘wants’ in an essay rather than think in terms of what constitutes solid writing.
There are certainly many potential issues wrapped up into why students are giving numerical answers to this problem. It does appear that some students are overriding their common sense… and my hope for this video is that it will help us all reflect on why that is.
When I was a private tutor I would mess with kids’ heads and pretend they could do more work on a problem. They would always sneak looks to see if I was turning to the next problem, so I would keep on looking at their work waiting for them to do more. For the kids who had really mastered the material I would lie and tell them they could do more work, so they really had to get in my face. 🙂
The problem I see with this exercise is that there is an implicit contract between teacher and student that the teacher will ask questions with answers. What happens if one gives the same question and offers as possible answers 130, 120, 25, 625, other___ and “cannot be computed”? Oh, and 42 of course. 🙂
kenneth
You make an interesting point Kenneth. You are likely right, that for some reason, there is an unspoken obligation to deliver an answer when someone asks for you even if it seems nonsensical. I don’t know what would have happened if it was a multiple choice problem with “cannot be computed.” If I had to take a hunch, I think more people would have chosen “cannot be computed” than just 8.
I used this as a warm up with my 4th grade class today and it was pretty funny. Some of the answers I got were 130, 625 and 21. Most students wrote some sort of number sentence with the numbers and one spent a while trying to figure out if maybe the shepherds age had something to do with how many animals he acquired each year.
Two students wrote down, “I have no idea. . .this question makes no sense”. They thought it was hilarious that I said their answers were correct. They both felt like they had failed at the task.
My favorite answers were 21 and 625. The student who answered 21 explained that “21 seems like a reasonable age for a shepherd”
When the other student said 625, the other students called out that they didn’t think that was a reasonable answer since nobody can live to be 625 and he said, “well, i’m pretty sure there were some really old shepherds in the bible so it seemed reasonable to me”.
Teri, your stories crack me up. Thanks for sharing. Yes, I had similar explanations for both the guesses and the old shepherd ages, though that was for 120 and 130. Hopefully they will learn from this. I wonder what would happen if you asked them a similar question again later in the year….
I feel that the one who said 21 because that’s a reasonable age for a shepherd was basically just as right as the two who said they didn’t know the answer. Saying it’s a reasonable age implies that the answer cannot be computed.
Good point Kai.
actually he is correct
there are people in the bible who lived for undreds of years
kudos to him for knowing the bible!
First, what’s up with your timecode stamp on these posts? Ha! It’s December 3 at 9:30 pm and I see a post from you on December 4, at 3:41am. Go figure.
Thanks Robert for posting this and sharing. This blew my mind when I tested it out with my students. Here are the results:
7th grade classes:
1 of 23 students had the correct conclusion.
1 of 39 students had the correct conclusion.
6th grade class:
0 of 22 students had the correct conclusion.
We had a great laugh looking at each other’s conclusions on the document camera. I think many of my students learned a valuable lesson: don’t think that every question has an answer simply by performing one operation with two numbers. Thanks for recording, editing, and posting this, Robert.
Keep up the great work.
LOL. Ok, just figured out how to set my website’s timezone… so I think I am going back in time to respond to your comment before you posted it. Thanks Andrew for sharing all the great data. I am curious, as with Teri, if students learn from this. If you try a similar question later on in the year, I hope they do better! Definitely makes you reflect on what is causing them to reach their conclusions.
So, how old *is* the shepherd?
I thought this was about the Sheep market scam, thanks for tricking me into reading this instead dude!
This is case in point for a few problems:
1. When given a question, people feel the need to answer it. “i dont know” isnt socially acceptable anymore, school’s curriculum’s deal only in the knowable, not the unknowable.
2. When we base schooling around testing, students learn to look like they’re smart in order to fool the test. So many of those students conjured some work to try and get a mark or an answer without understanding what it was they were doing. The probably have had to do lots of maths without understanding the words, just transposing numbers into equations.
3. A school system that focuses so heavily on Math (and science) makes it difficult to learn lateral thinking like “Does this make sense? What is the meaning? How does this fit into the big picture? etc” Math and science tend to have modernistic solitary absolute answers , whereas the latter tends to have romantic modernistic opinionated answers.
MG, I don’t agree with your 3rd point. I don’t believe that “a school system that focuses so heavily on Math (and science) makes it difficult to learn lateral thinking like “Does this make sense? What is the meaning? How does this fit into the big picture? etc”
Take a look at some of my lessons and I think you will see where math curriculum intersects making sense of mathematics.
My high school geometry students laughed when they saw the problem. When I told them they couldn’t hand in a blank sheet, though, about half of them still gave me 25. The majority of the rest tried to guess a resonable age or put a joking answer.
My school-within-a-school alternative students (~Alg. 2 level) were split. Of the 6 who came to class today, I got 3 25s and 3 variations on impossible (including one null set).
Paul. this is a really interesting problem that seems to be far more common than I realized. The fact that they can laugh at the question yet still answer 25 when they felt accountable to turn something in is a problem we all need to reflect on improving. Also, I love your null set answer from a perspective that it is certainly better than a random guess, though the shepherd certainly had an age, we just don’t know it.
It was interesting; the student actually wrote “Not enough information ∅”. To him (I conjuecture) the symbol is synonmyous with “this problem can’t be solved” rather than “no possible solution can exist”, which is more of an important distinction than I had ever really considered before.
That’s great. If nothing else, I hope that this whole experience gets us to reflect on our practice. Thank you again for sharing.
I think this is a bit unfair. Sure, to us adults the question is absurd. But young people are learning, and they trust adults to teach them something. They are groping for a possible new relationship between sheep, dogs and the shepherd. Maybe there is a rule that says that shepherds accumulate N sheep per year? The bigger the flock, the longer the shepherd has been in business?
Students are conditioned to assume that the teacher is teaching something sensible. They don’t expect tricks, and are trying to please.
Pierre, the other day I had to laugh at myself because I was asked a question that I realized had an error in it, but instead of inquiring about the question, I assumed it was a mistake and answered it anyway. Basically, I fell into the same trap I am talking about here.
Your point about the shepherd rule would have been really interesting but no one brought it up. Most of the students explained their reasoning for dividing along the lines of the last student in the video.
The question now is how do we improve this reality?
Amen. (jumping in because this was linked to by a math coach today) I strongly suspect that some of the students will have had teachers they were smarter than before, and learned the consequences of challenging, and gave “an answer” knowing full well it was ridiculous… but teachers are stupid anyway so you play the game.
Many others though, will trust that there *is* something behind the question. I hear math teachers talk about the value of struggle. So as a student I’m supposed to trust you that you haven’t given me an impossible question, and “struggle” with it.
If handled carefully, I could see this building autonomy in some students, especially if there are other opportunities for that. However, I could also imagine students simply feeling that their judgment that school is about a power struggle and the teacher always proving you wrong had been validated. (Those are the folks who come to my office at the community college…)
Thanks for chiming in. There are definitely a variety of potential causes as to why this is happening. Personally, I’ve come to see it as a matter of intellectual autonomy, but that could be connected to earlier life experiences like you describe. Thanks again for forcing me to reflect on my own reasoning.
Thanks for posting–while I am not an active math teacher, I appreciate you exposing the very predicaments that these children face in a world so hyper-focused on ‘mathematics’. I recently came across a wonderful, short essay (now book) called A Mathematician’s Lament by Paul Lockhart that argues the very problem in our math classes is the lack of math (the art of thinking, creating, and finding meaning). He provides some wonderful ways in which he provides students to actively engage with their own intuition, such as letting students play, ask questions, get frustrated, create, explore, etc. to discovery that math is simple and beautiful (p24). He also suggests that teachers need to be real and to encourage students to be real rather than to play the game. “Of course, it is far easier to test someone’s knowledge of a pointless definition than to inspire them to create something beautiful and to find their own meaning.” (58) Re: geometry, proofs and theorems, he shares his opinion and perspective that definitions need to come out of problems not the other way around, which can also be said for how students think in general about problems and how to approach them, “The problem with the standard geometry curriculum is that the private, personal experience of being a struggling artist has virtually been eliminated…The result is that the student becomes a passive participant in the creative act…They are being trained to ape arguments, not to _intend_ them. So not only do they have no idea what their teacher is saying, _they_have_no_idea_what_they_themselves_are_saying_. Even the traditional way in which definitions are presented is a life. In an effort to create an illusion of clarity before embarking on the typical cascade of propositions and theorems, a set of definitions is provided so that statements and their proofs can be made as succinct as possible. On the surface this seems fairly innocuous; why not make the abbreviations so that things can be said more economically? The problem is that definitions matter. They come from aesthetic decisions about what distinctions you as an artist consider important. And they are _problem_generated_. To make a definition is to highlight and call attention to a feature or structural property. Historically this comes out of working on a problem, not as a prelude to it.” (p78-9). In other words, to help students be real learners, we teachers must also be learners and find our own excitement in math, give ourselves permission to let our students play and come to their own answers, provide time and space for students to engage with problem-solving (which inadvertently goes against pressures from government, school and parents), and be willing to risk our lives (or jobs) actually teaching to help awaken the student for what they are up against with the conditioning and training students to perform rather than to explore, play, and embody learning so that creativity and ingenuity flow forth.
That sounds like a very interesting read, Jessica. Thanks for sharing. Yes, students certainly need more opportunities to make sense of mathematics. I hope that some of the real-world problems I love so much will give students those opportunities.
I think Getting a problem wrong and then learning why is more valuable than getting it right. The students that failed this problem probably learned a lot in the explanation and discussion about problem solving and critical thinking. My middle school had a trimester-long course in sixth grade that dealt entirely with this type of material.
My favorite challenge was when our teacher asked us to draw a bicycle. No one drew an accurate sketch of a bicycle, even though we all owned one, or had at least ridden on one. We all drew two tires, some included handle bars, seats, peddles or chains, and usually straight lines connecting everything together. They all looked like the doodles of a toddler. We all laughed at our silly drawings when the teacher showed a simple sketch of a bike along with a photograph of one. It wasn’t any more detailed than ours, but it had the lines of the frame and all the parts in the right places. This was the first time any of us had thought that a bicycle wasn’t just two tires and some other parts, but rather those parts were all connected in a logical way that made them work together. If you don’t think about how the parts work together, you can’t draw a bicycle any better than a 3 year old.
Hopefully, all these students seeing the shepherd problem learn that you can’t just manipulate information to get an answer, you need to make sure you have the right information first.
Dan, I completely agree with your first statement of “I think Getting a problem wrong and then learning why is more valuable than getting it right.” I think asking students this question (or a similar one) and then having conversations about it may lead to a very valuable self reflection for all involved.
Interesting what you were saying about a “rule.” I’m a math teacher, but I just asked my own kid. She’s 7, and her response was to stare at me, laugh, and then ask for more information: “Is there a year for each animal or something?”
Dividing wouldn’t have occurred to her, though, since she doesn’t know how.
I like her response; I just hope our overtesting world doesn’t teach that right out of her.
That’s a pretty good response for any age! Yeah, it will definitely be an interesting experiment for you to occasionally ask questions like that to your daughter.
This is fascinating and horrifying. I do wonder about how many of the kids who said it didn’t make sense would say the same thing for a problem that actually DID make sense, but took a bit of thought. I’ve been having that problem with my students for the last few years.
They see sense where there is none AND the other way around.
You make a great point Justin. It makes me wonder what that question would be to ask students that DID make sense. Any thoughts?
I recently gave a test from a new textbook. Question 12 asked if the graph was linear or non-linear. About half got it wrong. While helping a student with this same test, I saw that the student edition of the book had no graph. They all guessed, and not one told me there was no graph. The only student who made the least bit of sense was one that said it was non-linear because there was no line. I am going to try this question on the next.
That is a fantastically horrible story. I guess that shows you that given two random choices, half will pick the correct one. Let me know how this goes!
On the odd chance it was a trick question with a real answer, I reasoned that I might be the Shepherd, and gave my own age as the answer (28). The assumption that mislead me was assuming that the question had a correct answer.
Even this would be a more reasonable explanation. Well played.
Really interesting stuff Robert! I can’t wait to give this question to my two classes of 8th graders. I’m guessing my results will be similar to the ones you found but we’ll have to wait and see. I honestly think that students have been so conditioned over the years that every question not only has a “right” way to solve it, but a “right” answer as well. They’re used to figuring out what question the teacher wants them to answer and then trying to shoehorn whatever information they have into some kind of formula or calculation to find the answer. I think having students “making sense” of problems is one of the best ways to gauge their problem solving skills. You can tell a lot about them when you give them an open ended question with no clear path to an answer. I find that the more I do that with my students, the better they are getting at it. It’s really great to see. Love you blog posts! Keep up the great work Robert!
Thank you John. You make lots of good points. I mentioned this in another reply but the other day I had to laugh at myself because I was asked a question that I realized had an error in it, but instead of inquiring about the question, I assumed it was a mistake and answered it anyway. The error in the question was intentional to make a point about people’s listening skills and basically, I fell into the same trap I am talking about here. So, it makes me wonder whether the scope of the issue is the American classroom, American society, so something even bigger.
Let me know how it goes in your classroom.
My first response was “isn’t it a herd” of sheep not a flock.
My second response is I miss the Glen Rock Galley and need to make an East Coast comeback in 2014-15 even if I’d still waste my 2nd round pick on Torry Holt.
My wife’s response is “how much is this a mathematical issue and how much of it is a personal confidence issue”? How many of the kids were quietly questioning your illogical question and wanted to say “Mr Kaplinsky are you an idiot” but were afraid to argue against an authority figure? Was there a percentage of students who preferred to give an incorrect numerical question rather than argue the question itself?
I remember taking a adult-ed statistics class and got in a lengthy argument with the teacher over the Monty Hall problem (http://en.wikipedia.org/wiki/Monty_Hall_problem). He insisted the odds were 50:50 whereas I argued the 33:67 percentage because it wasnt a true statistical problem. I lost the argument that day in an extremely lonely classroom but still got an A in the class. I do remember though that even as confident math nerd as I was as a 22 year old, it took a great deal of guys/moxie to argue with a college professor and half way though I was questioning myself for even raising my hand in the first place.
Great question and an absolutely lovely presentation! I wish all math teachers took their profession this seriously as you!!!
PS I hope someone is beating both you and Dan in the standing…
Hey Marc! Good to hear from you. Your wife makes a very valid point that honestly I had not considered prior to filming and posting this video. Clearly, some component of making sense of problems is the possibility that you can question the question itself.
I remember reading about the Choose a Door problem. It is a great example of something where the correct answer seems counterintuitive. Your story is also a good example of how critical thinking can appear to be questioning authority to some.
Thank you for sharing and it is great to hear from you!
I see a lot of posts questioning the validity of the question due to the authority asking for an answer, and others relating that people feel pressured to provide an answer (even in cases where they know it is a bad question to begin with).
I wonder if the results would be different if the students were broken into small groups of 3-4 and asked to solve the problem collectively. This might provide courage enough to overcome the influence of authority. Also – considering 25% tends to get this right and each group would probably have someone who would solve it correctly – I wonder if that individual would be convincing to the group or if some sort of group quest for an answer would come into play.
I went online to see if anyone had done that experiment and came up blank which surprised me.
You make some great points. As people add more comments, I think of other ways I would love to modify the scenario. For example:
– What if we changed the context?
– What if we made it multiple choice?
– What if after we asked the question we said, “Another student said that answer was 35. How can we tell if that student is correct?”
Small groups is another variation. For what it’s worth, I was also surprised prior to filming that I couldn’t find any other videos online.
What a fascinating post and set of responses. No longer a practising teacher, my methods were in mathematics and computer science, I am always interested in current practice.
My first thought was that you can’t have five dogs in a flock — it should be a pack! 🙂
The comments on the authority gradient are sound. Couldn’t this exercise be used as the basis for a discussion on this point with the students?
FInally, when I gave this problem to my two kids (ages 8 and 10), I got “fish” and “it doesn’t matter, as long as his chicken is hot”. Too much exposure to Monty Python and Dali, I suspect… but did anyone get any surreal comebacks when they tried this on a larger more heterogeneous sample set (I mean group)?
Sounds like your kids have a witty sense of humor. Thanks for sharing!
Ha-ha! But do YOU know how old is the shepherd? Where do you know it from?
Actually, even the teacher doesn’t know how old is the shepherd, so how can he verify if an answer is correct or not?
Joke aside, the guessing is not that bad.
If you suppose that a shepherd is between 1-100 years old, with guessing, you have a chance of 1%.
Not too big, but still better than not answering anything (which means 0% chance of getting it right)
Of course, the best answer is: there is not enough information.
In high school, I had a test where a GOOD answer was worth 1 points, but a BAD answer was worth -1 points!
So it was better NOT to answer a question if you were in doubt. The point was to keep out guessers, and only show knowledge.
NU, if kids were to make a guess and explain it as you did, I would be thrilled. Definitely a higher likelihood of getting it right!
My 8-year-old homeschooler: Huh? …I don’t GET it.
Me: Why not? What’s the answer?
8-year-old: This problem is unsolvable.
I should add that her math curriculum sometimes throws in problems that can’t be solved with the information given, and sometimes provides extraneous information that isn’t needed to solve the problem. So she didn’t come in with the assumption of the perfectly reliable teacher.
This is great Rebecca. Sound like your homeschooler has a good head on his/her shoulders.
What I wonder is, were the students debriefed afterwards, so that they could learn from the experience rather than just serve as data points for how our teaching of math is broken?
And if they were debriefed, how did they react? What did they take away (or not take away) from this?
Janni, the students that were debriefed had a good laugh at the problem and the answers. I hope that their takeaway was that they should trust their judgement more. That being said, it would have been nice to have them write out a reflection for next time. Thanks.
I tried this on my seven year old daughter. First she asked if she was supposed to add or subtract. I answered with “That’s not what I asked. I asked how old the shepherd was.” That got me a frustrated scream of “I don’t know!” so I told her it was the right answer.
Yes Therese, it appears that many people do not know how to cope with the problem’s nonsense. Some laugh at it. Other get frustrated, quiet, or even embarrassed. Definitely interesting to watch it unfold.
I would say that the student who answered “42” answered correctly, and that you were asking the wrong question.
Hahaha. Well played.
I asked my kids — one is seven, the other is ten — and the seven-year-old said, “130! No, wait…wait…” and his sister said, “No, mom, that makes no sense!”
I think that anyone who can work five dogs simultaneously is very accomplished, no matter what their age. 😉
…and then I asked my husband, and he tells me he needs to know how many wives the shepherd has before he can answer.
Ha! Thanks for sharing. It is fun to experiment on our loved ones.
I think the context plays part. Asking this question on a math class, English class or a philosophy class could yield different results. Let alone asking this at home from your own children.
Certainly you are right. My hope for sharing this video was to get people thinking about this and reflecting on opportunities for improvement.
This is fascinating, and reminds me of a continuing argument between some of the students in my (college undergrad) lit class and our professor.
He will, occasionally, ask a question on an exam whose answer cannot actually be derived from the source material. Either the question is subjective, or the source is contradictory or unclear. These aren’t phrased as interpretive questions, and there’s no room to add an explanation; the questions are often multiple choice, and some have even been strictly binary. When does character Z do A? (He doesn’t, actually, or it depends on how you define A, and even if you define A this certain way you could make equally valid arguments that A occurred in Acts 1, 3, or 4.) Does character X believe Y? Answer Yes or No. (We’re never told, there’s no subtext indicating X’s beliefs on the matters, and none of X’s actions can be attributed to whether or not they believe Y.)
I don’t think any of us have ever just written “I’m not answering this because the question is inherently flawed and unanswerable,” because we’ve been trained our entire lives to believe that isn’t an option. Instead, we write down our best guess as to what we think the professor wants to hear, and then spend the next class session arguing about it.
He doesn’t seem to understand why this turns the class into a mutinous, indignant froth.
(A few times I’ve asked him to show where in the text he found the answer he’s grading us by, and not once has the text has been definitive.)
Whenever the class complains, he just shrugs and says we should answer however we feel. If we believe the question is unanswerable, we should write that. If we believe there’s more than one answer, we should write both of them. “Just don’t overthink it,” he told us. (The girl next to me sighed and finally raised her hand: “So we don’t need to write a justification of our answers, unless we believe that our answer isn’t the answer we think you wanted, in which case we should clarify, but we’re not supposed to be concerned with what answer you want?”)
One day the professor got frustrated and asked us why we felt such a need to provide an answer we thought would please him. “Because we’re getting graded,” two people shot back.
Bonus interesting context: As far as I know, every student who’s complained about the subjectivity of the questions attended K-12 in the United States. Our professor is not from the US.
That really is an interesting story. I haven’t ever experienced something like that but I see how this video would remind you of it. I wonder what your professor’s answer key would look like…
I recently had my students do an exercise where they were given a following directions test. Here’s an example http://www.docstoc.com/docs/116302283/following-directions-timed-test but these things are all over the internet. I scaled mine for 1st grade and up. Out of 20+ students ages 5-12 not one of them figured it out (except the kid who finished first and then frantically tried to erase everything!). I’m always getting on them (I tutor) for not completing homework correctly and making silly mistakes because of not following directions. We all (the students) had a good laugh when they realized what had happened!
I will have to try your shepherd test and see what I get. I bet they will love it.
It seems that most kids (even in kindergarten and 1st grade) have TONS of homework and simply dive into completing it so they end up with quantity not quality. Given a problem or as sheet of problems = ANSWER THEM ASAP!!!
It really frustrates me as an educator that I have kindergarteners who are crying and say their stomach hurts because they are scared and anxious about going to school. They tell me they HATE school because it is too hard and they have too much work. Their teachers are telling their parents that their child is FAILING KINDERGARTEN because they cannot READ (4 and 5 letter words) before the end of the fall semester. *pulls out hair*
I tested my sixth graders!
18 out of 54 stated there was not enough information.
As we debriefed the experience, they commented that many of them initially thought there wasn’t enough information to solve the problem, but then second guessed themselves. They didn’t think that their teacher would intentionally trick them, so they figured there must be some solution. This lead to a great discussion on intellectual autonomy and intellectual courage.
Thanks for this great article! When students tackle problems, I now remind them of the shepherd and ask them to consider if their answer “makes sense”.
Wow Cari. That is definitely the highest percentage (~33%) of correct answers from a large group of students that I have heard yet. I am so glad it led to such an important conversation. That is definitely an outcome I had hoped might happen from sharing this video.
A colleague asked his 14 year old son the shepherd question. His son’s response? “That’s like saying, here’s five apples and three bananas. What’s the diameter of the sun?”
We wish we had a video!
Love this post. Powerful and provocative.
Ha! I asked my kindergartener a version of this question about a boy who has 3 dogs and two cats. He had no problem saying that he was 5 years old. Hopefully that will change over time. 😛
I would like to see a lesson that consists entirely of word problems, each of which contains twice as much data as would be necessary, and half of which do not contain all of the right data to answer the question posed, for which the instructions read “Answer the questions which can be answered with the information given. For those which cannot be answered, indicate a piece of information, which, if known, would allow the problem to be solved. Using your own experience, give a reasonable estimate of this quantity and solve the problem using your estimate as if it were the correct value. Give correct units for all answers.”
Have you ever seen a worksheet of this sort?
Interesting idea. The closest thing I have heard to what you describe (and is not close at all) is something like this: http://blogs.scholastic.com/files/followdirection.pdf
It would be fun to give the same test to a group of teachers.
My guess would be that a large percentage (maybe not 75%) would also answer with a number guess.
This is a really hard problem to give to teachers. I’ve done in numerous times, but inevitably someone starts giggling and it basically ruins the experiment.
So I know I’m really late to this party, but I asked this of my own three children (1st, 3rd, and 6th). Each of them responded with an initial look of confusion and some type of exasperated “it is impossible” response. Sadly when I asked them why it was impossible each of them immediately changed course and gave me 130 as an answer. I know your focus was MP#1, but it turned in to a great discussion MP#3 discussion. They lost confidence when I questioned them, and instead of trying to justify their response they immediately fell into shoving numbers around. I wish I could find a way to bottle math confidence… I think i could make a fortune!
Thanks for sharing this Bob. I’ve had similar experiences when talking with other children. I hope this gave you a great opportunity to talk about this with your kids and at least bottle a bit of confidence for them.
“…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.”
Wouldn’t that be 30 years?
Hi Jeanne. Thanks for checking my math. If you look at the whole paragraph it says:
“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.”
This blog post was written in 2013 and I mention that Professor Merseth wrote her essay in 1993. 2013 – 1993 = 20 years.
When all the information to do the problem is not there it is human nature to fill the void. We ,as humans will always want a resolution even when it is not possible.
Thanks Karen. We will have our work cut out to help people supress this desire long enough to make sense of what they are doing.
It would be interesting if you added some words that teachers tell students indicate addition or subtraction, etc. I am having so many issues with my 4th graders who keep telling me that if ‘more’ appears in the problem we have to add!!
Kim
Can you tell me more about what you mean Kim? I’m not sure I understand.
One of my colleagues has a wonderful story. He was observing a 4th-grade science classroom as the teacher walked around with a metal pan, some sand, and some water flowing down the sand forming a “delta” at the bottom. The teacher showed it to a student and asked, “What does this show?” The student obediently responded, “Erosion and deposition.” When the teacher walked away, my colleague asked the student, “What are erosion and deposition?” The student shrugged his shoulders. The student had no idea what the words meant. My friend coined the term, “vocabulary self-defense.” The student had learned that when the pan of sand and water shows up, if he says, “erosion and deposition,” the teacher will go away and leave him alone.
I don’t know what I like most about this comment: “vocabulary self-defense” or the sad reality that this is a strong example of the same thing happening outside of math. Did either of you bring it up to the teacher afterwards? If so, how did the teacher respond? What suggestions would you give the teacher as a result of that experience?
Yes. My friend had a conversation with the teacher afterward about good vocabulary strategies. Our approach/theory to vocabulary is that the term itself (erosion, deposition, revolution, etc) should be given to the student only after the student has described the occurrence. So, as an example, if the erosion and deposition concept were being introduced, the teacher would show the students the water flowing in the pan (or better yet the student would be doing it themselves). The teacher would then ask, “Describe to me what is going on in the pan.” The student would reply with a description, “Well…the water is flowing down and moving the sand with it.” Only after that would the teacher say, “In science, there is a word for that. Its called erosion. Now…what is it erosion?” The student would then re-explain the whole thing. The idea is that a student has experience and understanding of the concept first, then the label (vocabulary word) is simply hooked onto the idea that the student already understands.
Wonderful. Thanks for sharing all of this. I try to do the same thing in math, almost down to the wording, “Other mathematicians have also noticed this and they call it __________.” Great stuff.
Cari’s idea of intellectual autonomy and Kenneth’s idea about that implicit contract between teachers and students has been on my mind lately since I have been introducing many teachers to open questions. I am an Instructional Coach and I have been using Marian Small’s open questions for years in my own classrooms; really love @OpenMiddle too. What I find is that children are stymied when faced with questions that have more than one possible answer. We have conditioned children to look for THE ONE RIGHT answer. I don’t think many educators would argue with that. Changing this mindset and encouraging children to think for themselves, have confidence in their capabilities, and take risks is a long term project, in my humble opinion.
Crazy story to tell about what happened TODAY. I picked up my daughter who is in grade 4. She was on the verge of tears because, as she was leaving the building, a teacher told her she was “too tall to be in grade 4 and should be exiting the building at the other doors with the kids in grade 5 and up”. Rather than explain to the teacher that she was mistaken, my daughter turned around and did as she was told. My point is that there is no way she would ever tell a teacher that his/her math problem didn’t make sense (a teacher wouldn’t break the contract!!). And, she would stress about the fact that she couldn’t figure it out. I want to give students lots of opportunities to make sense of the world through math and not see me as the guardian of mathematical knowledge. Lots of work to do here. Thanks for the post, Robert!! Will be sharing.
That’s a hard story for a parent to learn about, I’m sure. We all hope that our students, and especially our children, will have enough confidence in themselves that they can overcome those moments when people of positional superiority seem to be posing a dilemma. I do appreciate this story and look forward to learning any insights you pick up along the way.
I gave this question to my 3rd grade students 24 years ago when I began my teaching career to prove the article I had read on it, wrong! Sadly, I found my students respond in a similar fashion like the students in the study. What is even more sad and somewhat perplexing that 24 years later, making sense of math word problems has not sunk into our educational system. Why?
Wow Satinder. You shared a very sobering perspective on all of this. It is definitely worth questioning why this has not improved over the decades. I have my perspectives but I’m not sure whether I’m right. Here’s a short version of my thinking: http://robertkaplinsky.com/is-problem-solving-complex-or-complicated/
There seems to be a lot questions regarding the “lack of knowledge of the context” or the “fear of not making the teacher happy” comments. I once ‘tested’ my 3rd-5th grade students in a similar fashion. I continuously ‘accused’ them of not reading problems before they tried to answer and they would always assure me that they had. So I prepared a paragraph that included a story about a context with which they were familiar. Some numbers I wrote out in word form, others in numeral form. Their size varied between 1 and 2 digits. But embedded within I story were sentences like: The answer to numbers 2 and four is “a”. The answer to numbers three and 1 is “c”. Four multiple choice questions were at the bottom of the page. Of the 50+ students who were given this problem that day, only 2 got all 4 answers correct. As I read the problem out loud afterwards, there were always groans and “that’s not fair” in the class. Some actually began reading and trying to make sense of the problem. I even heard a couple say, “I don’t trust her. This might be another trick. I’m gonna read it.”
Thanks for sharing this story Robyn. I think it speaks volumes to how meaningful word problems are to students… which is to say that they are fairly meaningless.
This is why I prefer the problem-based lessons I share under “Lessons”. I find that students are more likely to make sense of them, and they more closely align to real life as well.
Hi Robert,
I find it interesting that there was such a high number that came up with an answer. My 8th graders tried to answer the question too.
My question to you is why are you showing this to the parents? To me it does not show anything positive about math today, but rather the opposite.
My question is why are you concerned about showing this to parents? This is our reality. Parents often wonder why we are putting greater emphasis on sense making and I think this does a good job of starting a conversation about answer getting versus sense making.
I think the video does make a good point to parents that students need more than just computational ability. However, Paul makes a good point that with so many current examples of students struggling with both the computation and the understanding that computation is pointless for this pseudo-problem, parents’ doubts about process standards may be reinforced. They could logically reason that “Back in the good old days, we might also have missed this but at least our computations were accurate.” For me, it reinforces the need for us to get both tasks done.
It’s worth noting that this video was filmed prior to Common Core being rolled out in California. I think it’s also worth examining how people pull out the meaning they want from a context.
I know this is 75 because i have a much stronger reason. In the simple word to i explain. The question is about to what is the age. There is the age means experience not the real age. Because in a flock there is totally 130 animals sheeps and dogs but the thing is these animals not can controlled by 1 person. Normally required person is to be 5 to controll then it means 5 people controlling this. This people are all called shepherd. So each person age is include in question means by the combination of 5 persons experience make the age and the average age for controlling sheeps has to be 15 years old atleast this all persons experience make the age 75.
I teach adults a beginning Math class. Some of my students went right to the math, adding or subtracting. Some kept asking questions, and some just got pissed because they said there was no way to figure it out. Contradicting the teacher is not a problem-ever! We got some good laughs. We got even bigger laughs when they watched the video and some figured they were ‘right’ – this is impossible to answer. Thumbs up on the question!
Awesome Ruth! I’m glad it turned into a good conversation for everyone.
My 6th graders replicated the strategies mention previously with one exception. One student took the word “shepherd” to refer to one of the dogs. So the solution was 13, because that’s how long sheep herding dogs typically live.
Hey, that’s a lot more reasonable than most!
This was really fun for my students. I did it with my 6th, 7th and 8th grade classes this morning for our shortened “Remote Wednesday” schedule. I got a lot of interesting answers, and some really funny ones, but what I found most interesting is that in 8th grade, 25% of students gave me a numerical answer. In 7th grade, 35% of students gave a numerical answer. In 6th grade, 83% gave a numerical answer. I wasn’t expecting that, but it shows a progression to more critical thinking as they get older.
That really is unexpected. For perspective, the students in the video are eighth graders. I wonder if your colleagues could do this with their kids and see if the pattern continues. Then figure out what happened after or during 6th grade and let me know!
It could also show that your school is doing a good job of promoting the thinking skills in math.
I just found your video. Wow! I asked my 10 and 12 year olds the question. Thankfully they both said you can’t do it.
Robert, looking forward to seeing you in Sydney very soon.