Yesterday 33 boats sailed into the port and 54 boats left it. Yesterday at noon there were 40 boats in the port. How many boats were yesterday evening still in the port?

I recognize this framing as the kind of “standardized test” question that you were expected to “fill in the blanks” on when I went through school. Given such a problem, an answer is expected even when the information contained in the question is too vague or incomplete to produce such an answer. If challenged, the most common result is to be marked wrong, told your reasoning was wrong and often also results in angry teachers. Often answering this type of question “correctly” would require looking at the multiple choice answers and deducing the proper framing from the available answers that seemed most correct.

Compare this to how a similar problem may be handled in a university mathematics education.

I remember on my linear algebra final, which was my last final of the quarter, that I forgot the method to solve one of the questions. After trying for some time to recall the mechanics of finding the solution, I opted to derive a best guess at how to get the answer, along with a short explanation of what I had done and what reasoning I used to derive that method. This answer was accepted for partial credit (I believe I got the answer slightly wrong, but was suitably close and provided sufficient explanation as to convince the professor that the problem was one of forgetfulness rather than lack of facility with the material).

The thing to note is that I understood myself to have the freedom to do that. The class was not such that I felt forced to use the method which had been prescribed, and it was clear the intent of the exam was not simply to get the right answer in the exact mechanical way the professor intended it to be had, but to illustrate that I had actually become literate in the language (the mathematical objects, axioms, theorems, and processes) of linear algebra.

intent of the exam was not simply to get the right answer in the exact mechanical way the professor intended it to be had, but to illustrate that I had actually become literate in the language

That was the core feature of my post-calc math courses in the process of my math minor.

God! With that many sheep and that many sheep dogs, obviously the shepherd is no beginner. However, with that many sheep and that many dogs, obviously the shepherd is a spry man. I’d say the shepherd is between 30 and 40.

that’s a lot of dogs for a flock that size. could be that the shepherd (as likely a woman as a man these days) is quite young (dogs there as backup) or older (training the extra dogs, dogs there as backup if 55+). I’d also argue that “spry” ends in the mid-50s. so we’re easily taking 12-75 here.

In many instances, the preference now is to prefer a single term for both genders, rather than using gendered variants. For example, “waiter,” “host”, or “server” for both male and female waitstaff at restaurants, or “flight attendant” rather than “steward” and “stewardess” on planes.

Nearly all the standardized tests I took had an explicit option for “not enough information to solve the problem” for questions exactly like this. Did somebody eliminate the answer but forget to remove the question?

It seems what happened is that the teacher intended this question to be an opportunity for students to show their facility with the subject by proving that no answer exists, except this intention was never stated to the class, and the problem was never reviewed to ensure such understanding. The parent and child in question only found out because they reached out to the teacher on their own.

Given this, it seems like a decent idea with terrible execution, which isn’t terribly surprising. K-12 mathematics education in the US seems never to be concerned with mathematical literacy and is instead entirely interested in “getting the right answer,” with the implicit lesson that such an answer always exists.

K-12 mathematics education in the US seems never to be concerned with mathematical literacy and is instead entirely interested in “getting the right answer,” with the implicit lesson that such an answer always exists.

While I agree, I believe this was the UK public school system. ;)

It was, I just don’t have any experience with the UK public school system, and so I didn’t want to speak to what it is like. I imagine it is similar to what we have in the US in this respect, but it may not be. If anyone has any thoughts they would be much appreciated!

I wonder if this is more an issue with the didactical contract [1] rather than lack of mathematical skills on the part of the students. In Brousseau’s view, teachers and students are bound together in the classroom by reciprocal responsibilities. Among them, the teacher has the role of asking questions and the students have the role of coming up with the answers.

The teacher breaks his/her contractual obligation by posing a question like the shepherd problem. Some students will call him/her out on it recognizing his/her contractual breach, while others will think that the question is some sort of trick question (thus working within the agreed-upon contractual roles of teacher and student) and will try to provide an answer to the problem.

As a math teacher, you are between a rock and a hard place. On one hand, you have to provide problems to your students to practice what they are learning in a way that doesn’t confuse them. On the other, you want them to develop the intellectual autonomy and problem solving skills that Mubeen talks about.

[1] Brousseau, G. (1997). Theory of didactical situations in mathematics: Didactique des mathématiques 1970–1990 (N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield, Eds. and Trans.). Dordrecht: Kluwer.

The answer then would be to make explicit the nature of mathematics. K-12 education in the US does students a disservice by providing a view of mathematics as being purely axiomatic, hiding away the discovery, inconsistency, and creativity of professional mathematics. When you learn math in school, you’re simply told “these are the facts” (the axioms and theorems of the field), with no indication of the reason of those facts, and the ways in which they were derived.

Compare that to the system illustrated by Paul Lockhart in his book “Measurement,” wherein you would give first the description of some mathematical objects (say, triangles), and then begin to ask and answer questions about what rules these objects obey based on the definition of their structure.

It may be that Lockhart’s free-form method is ill-suited on its own to testing-based education (we’ll leave the question of whether testing-based education is a good idea for later), but it seems entirely possible to meld this method with the more axiomatic structure of current math education. Start with an interrogation of the core objects in the area being studied, and then talk about what facts have already been discovered by mathematicians about these objects (perhaps giving students opportunities to derive these facts independently), and then using those pre-developed facts to further encourage exploration of the subject.

Unfortunately, to discourage guessing, certain examinations do impose a penalty for incorrect answers (I believe the AP exams do this, but it’s been a while since I last took one, and the policy may have changed).

It’s not to discourage guessing, it’s to keep things neutral. If you answer C for every question without even reading it, an accurate assessment of your demonstrated knowledge is 0%, not 25%.

i remember running into a teacher like that in eighth grade biology. he had messed up in one of his tests by asking for a diagram he had not yet covered in class, and when that was pointed out he agreed that it was his mistake, but rather than drop the question he decided to give full points to anyone who had drawn anything, even a circle, but zero to anyone who had just skipped the question. (i probably still recall the incident because i was one of the people who got the short end of that stick :))

When I was in grade four, we were learning how to round. My teacher told us that 5 and up rounds up, but she made the mistake of thinking this was recursive. For example, 3.4452 -> 3.445 -> 3.45 -> 3.5 -> 4. I think the error stemmed from how we were learning that you could round to a different decimal position.

I knew this was wrong. This didn’t make sense. However, I was also unable to explain why this was wrong. I yelled, I screamed. I was in tears, throwing a childish tantrum, trying to explain to the teacher why her method of rounding made no sense.

I’m not sure what my purpose is in telling this story other than to say that teachers make mistakes and not all children take mathematics as some adult-given immutable truth. Or perhaps I want to highlight how I was unable then as I am now to persuade others, even in the face of raw facts.

It’s certainly an example of how children get trained to follow directions above all else, in a school context. What especially bothers me is when I see teachers expressing surprise that their students don’t think independently. Well, yeah! The entire system is set up to discourage it.

Or the students just made a simple reasoning based on statistical facts:

“All the mathematical questions I have been given by adults had a correct answer. Additionally, all the questions I have received from them involved some combinations of +,-,*,/ operations. Therefore, it is HIGHLY LIKELY that this question also obeys those two criteria.”

And this reasoning would work in the context of school for years, it only fails when the students are deliberately given a trick question.

If anything we should be proud that the children have come up, all by themselves, with a very efficient way to reason about the solvability of a problem. I doubt that if the children were to happen upon a shepherd in the field, tending his flock of sheep, they would try to count the number of sheep and dogs, in order to calculate the shepherd’s age.

This is the point of the article. It’s not that the students are wrong; it’s that we’ve trained them to do this (for some primary education definition of “we”), and thus we’re at fault. It’s not about kids, except as their faulty reasoning being a symptom of our faulty teaching.

If anything we should be proud that the children have come up, all by themselves, with a very efficient way to reason about the solvability of a problem.

I’m not so sure. They’ve come up with a well-reasoned, but ultimately misguided (and potentially dangerous) way of dealing with the presented facts.

Now imagine these kids grow up and go into finance, or government, or middle-management with this flawed reasoning. In fact, that may well explain some of the observed incompetence and magical thinking in those fields.

In problem solving, everything, including the habits of teachers, is a clue.

Teachers do not usually ask insoluble questions and they usually ask questions that fit into the time period allotted.

These are valid clues to your problem solving.

The best students (as defined by those with highest marks) are very good at using these “teacher clues”.

There are many problems a struggling pupil will meet that are in the category… Gee, I don’t even understand the question. I must be dumb, or didn’t study enough, or ….

However I can use the contextual clues to make an… ahhh…. educated guess. Only one mark for this problem? Must be trivial. Only 1 minute assigned to solving this. Must be trivial.

Do something trivial that vaguely fits and move on hunting easier sources of marks, otherwise you will run out of time to do the questions you do understand.

This isn’t stupid behaviour. This is in fact smart behaviour.

So why do we even have word problems? They’re so dull, and they aren’t contextual and real-world!

Solving a real-world math problem (“is this shepherd an appropriate age to be dating my brother?”) is a very complex task. There are many phases - (1) articulating what you want to find out, (2) noticing all the evidence that you have available, (3) choosing the evidence sources that seem useful, (4) articulating the operations that will be used to combine the relevant evidence, (5) executing the operations, (6) interpreting the result, and (7) checking the result for reasonableness. Math teachers have to find ways to drill their students on each phase in isolation. Otherwise, someone who is not good at (e.g.) noticing all the evidence that’s available will never get any practice at the later phases, like executing the operations - they’ll be stuck, fail all their questions, decide they suck at math, and avoid it for the rest of their lives. Math is hard, let’s go shopping.

So I don’t think the issue the article raises is particularly damning of our education system, but of the experimental method. It looks like a question that a math teacher has set up to drill phases 4, 5, and 7, and the students actually did a pretty good job of applying 4, 5, and 7. But it’s a really a trick question and tests phase 3. If you really want to evaluate students' grasp of phase 3, then present the same information in the context of someone’s Facebook profile, along with lots of other useless information. It will never occur to them to divide the number of sheep by the number of dogs to find the shepherd’s age. Instead they’ll look at the profile picture, or see how old the person’s name sounds (Agnes? Madison? Wilbur?).

Basically, I am not impressed that someone is smart enough to trick a class full of 6th graders.

I recognize this framing as the kind of “standardized test” question that you were expected to “fill in the blanks” on when I went through school. Given such a problem, an answer is expected even when the information contained in the question is too vague or incomplete to produce such an answer. If challenged, the most common result is to be marked wrong, told your reasoning was wrong and often also results in angry teachers. Often answering this type of question “correctly” would require looking at the multiple choice answers and deducing the proper framing from the available answers that seemed most correct.

Compare this to how a similar problem may be handled in a university mathematics education.

I remember on my linear algebra final, which was my last final of the quarter, that I forgot the method to solve one of the questions. After trying for some time to recall the mechanics of finding the solution, I opted to derive a best guess at how to get the answer, along with a short explanation of what I had done and what reasoning I used to derive that method. This answer was accepted for partial credit (I believe I got the answer slightly wrong, but was suitably close and provided sufficient explanation as to convince the professor that the problem was one of forgetfulness rather than lack of facility with the material).

The thing to note is that I understood myself to have the freedom to do that. The class was not such that I felt forced to use the method which had been prescribed, and it was clear the intent of the exam was not simply to get the right answer in the exact mechanical way the professor intended it to be had, but to illustrate that I had actually become literate in the language (the mathematical objects, axioms, theorems, and processes) of linear algebra.

That’s cool, I had to do my own teaching for anything beyond basic calculus.

That was the core feature of my post-calc math courses in the process of my math minor.

God! With that many sheep and that many sheep dogs, obviously the shepherd is no beginner. However, with that many sheep and that many dogs, obviously the shepherd is a spry man. I’d say the shepherd is between 30 and 40.

that’s a lot of dogs for a flock that size. could be that the shepherd (as likely a woman as a man these days) is quite young (dogs there as backup) or older (training the extra dogs, dogs there as backup if 55+). I’d also argue that “spry” ends in the mid-50s. so we’re easily taking 12-75 here.

Isn’t shepherd male and shepherdess female?

In many instances, the preference now is to prefer a single term for both genders, rather than using gendered variants. For example, “waiter,” “host”, or “server” for both male and female waitstaff at restaurants, or “flight attendant” rather than “steward” and “stewardess” on planes.

Nearly all the standardized tests I took had an explicit option for “not enough information to solve the problem” for questions exactly like this. Did somebody eliminate the answer but forget to remove the question?

It seems what happened is that the teacher intended this question to be an opportunity for students to show their facility with the subject by proving that no answer exists, except this intention was never stated to the class, and the problem was never reviewed to ensure such understanding. The parent and child in question only found out because they reached out to the teacher on their own.

Given this, it seems like a decent idea with terrible execution, which isn’t terribly surprising. K-12 mathematics education in the US seems never to be concerned with mathematical literacy and is instead entirely interested in “getting the right answer,” with the implicit lesson that such an answer always exists.

While I agree, I believe this was the UK public school system. ;)

It was, I just don’t have any experience with the UK public school system, and so I didn’t want to speak to what it is like. I imagine it is similar to what we have in the US in this respect, but it may not be. If anyone has any thoughts they would be much appreciated!

I wonder if this is more an issue with the didactical contract [1] rather than lack of mathematical skills on the part of the students. In Brousseau’s view, teachers and students are bound together in the classroom by reciprocal responsibilities. Among them, the teacher has the role of asking questions and the students have the role of coming up with the answers.

The teacher breaks his/her contractual obligation by posing a question like the shepherd problem. Some students will call him/her out on it recognizing his/her contractual breach, while others will think that the question is some sort of trick question (thus working within the agreed-upon contractual roles of teacher and student) and will try to provide an answer to the problem.

As a math teacher, you are between a rock and a hard place. On one hand, you have to provide problems to your students to practice what they are learning in a way that doesn’t confuse them. On the other, you want them to develop the

intellectual autonomyandproblem solving skillsthat Mubeen talks about.[1] Brousseau, G. (1997). Theory of didactical situations in mathematics: Didactique des mathématiques 1970–1990 (N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield, Eds. and Trans.). Dordrecht: Kluwer.

The answer then would be to make explicit the nature of mathematics. K-12 education in the US does students a disservice by providing a view of mathematics as being purely axiomatic, hiding away the discovery, inconsistency, and creativity of professional mathematics. When you learn math in school, you’re simply told “these are the facts” (the axioms and theorems of the field), with no indication of the reason of those facts, and the ways in which they were derived.

Compare that to the system illustrated by Paul Lockhart in his book “Measurement,” wherein you would give first the description of some mathematical objects (say, triangles), and then begin to ask and answer questions about what rules these objects obey based on the definition of their structure.

It may be that Lockhart’s free-form method is ill-suited on its own to testing-based education (we’ll leave the question of whether testing-based education is a good idea for later), but it seems entirely possible to meld this method with the more axiomatic structure of current math education. Start with an interrogation of the core objects in the area being studied, and then talk about what facts have already been discovered by mathematicians about these objects (perhaps giving students opportunities to derive these facts independently), and then using those pre-developed facts to further encourage exploration of the subject.

[Comment removed by author]

Unfortunately, to discourage guessing, certain examinations do impose a penalty for incorrect answers (I believe the AP exams do this, but it’s been a while since I last took one, and the policy may have changed).

It’s not to discourage guessing, it’s to keep things neutral. If you answer C for every question without even reading it, an accurate assessment of your demonstrated knowledge is 0%, not 25%.

i remember running into a teacher like that in eighth grade biology. he had messed up in one of his tests by asking for a diagram he had not yet covered in class, and when that was pointed out he agreed that it was his mistake, but rather than drop the question he decided to give full points to anyone who had drawn

anything, even a circle, but zero to anyone who had just skipped the question. (i probably still recall the incident because i was one of the people who got the short end of that stick :))When I was in grade four, we were learning how to round. My teacher told us that 5 and up rounds up, but she made the mistake of thinking this was recursive. For example, 3.4452 -> 3.445 -> 3.45 -> 3.5 -> 4. I think the error stemmed from how we were learning that you could round to a different decimal position.

I knew this was wrong. This didn’t make sense. However, I was also unable to explain why this was wrong. I yelled, I screamed. I was in tears, throwing a childish tantrum, trying to explain to the teacher why her method of rounding made no sense.

I’m not sure what my purpose is in telling this story other than to say that teachers make mistakes and not all children take mathematics as some adult-given immutable truth. Or perhaps I want to highlight how I was unable then as I am now to persuade others, even in the face of raw facts.

It’s certainly an example of how children get trained to follow directions above all else, in a school context. What especially bothers me is when I see teachers expressing surprise that their students don’t think independently. Well, yeah! The entire system is set up to discourage it.

Ask a stupid question, get a stupid answer.

Or the students just made a simple reasoning based on statistical facts:

“All the mathematical questions I have been given by adults had a correct answer. Additionally, all the questions I have received from them involved some combinations of +,-,*,/ operations. Therefore, it is HIGHLY LIKELY that this question also obeys those two criteria.”

And this reasoning would work in the context of school for years, it only fails when the students are deliberately given a trick question.

If anything we should be proud that the children have come up, all by themselves, with a very efficient way to reason about the solvability of a problem. I doubt that if the children were to happen upon a shepherd in the field, tending his flock of sheep, they would try to count the number of sheep and dogs, in order to calculate the shepherd’s age.

This is the point of the article. It’s not that the students are wrong; it’s that

we’ve trained them to do this(for some primary education definition of “we”), and thuswe’re at fault. It’s not about kids, except as their faulty reasoning being a symptom of our faulty teaching.I’m not so sure. They’ve come up with a well-reasoned, but ultimately misguided (and potentially dangerous) way of dealing with the presented facts.

Now imagine these kids grow up and go into finance, or government, or middle-management with this flawed reasoning. In fact, that may well explain some of the observed incompetence and magical thinking in those fields.

In problem solving, everything, including the habits of teachers, is a clue.

Teachers do not usually ask insoluble questions and they usually ask questions that fit into the time period allotted.

These are valid clues to your problem solving.

The best students (as defined by those with highest marks) are very good at using these “teacher clues”.

There are many problems a struggling pupil will meet that are in the category… Gee, I don’t even understand the question. I must be dumb, or didn’t study enough, or ….

However I can use the contextual clues to make an… ahhh…. educated guess. Only one mark for this problem? Must be trivial. Only 1 minute assigned to solving this. Must be trivial.

Do something trivial that vaguely fits and move on hunting easier sources of marks, otherwise you will run out of time to do the questions you do understand.

This isn’t stupid behaviour. This is in fact smart behaviour.

So why do we even have word problems? They’re so dull, and they aren’t contextual and real-world!

Solving a real-world math problem (“is this shepherd an appropriate age to be dating my brother?”) is a very complex task. There are many phases - (1) articulating what you want to find out, (2) noticing all the evidence that you have available, (3) choosing the evidence sources that seem useful, (4) articulating the operations that will be used to combine the relevant evidence, (5) executing the operations, (6) interpreting the result, and (7) checking the result for reasonableness. Math teachers have to find ways to drill their students on each phase in isolation. Otherwise, someone who is not good at (e.g.) noticing all the evidence that’s available will never get any practice at the later phases, like executing the operations - they’ll be stuck, fail all their questions, decide they suck at math, and avoid it for the rest of their lives. Math is hard, let’s go shopping.

So I don’t think the issue the article raises is particularly damning of our education system, but of the experimental method. It looks like a question that a math teacher has set up to drill phases 4, 5, and 7, and the students actually did a pretty good job of applying 4, 5, and 7. But it’s a really a trick question and tests phase 3. If you really want to evaluate students' grasp of phase 3, then present the same information in the context of someone’s Facebook profile, along with lots of other useless information. It will

neveroccur to them to divide the number of sheep by the number of dogs to find the shepherd’s age. Instead they’ll look at the profile picture, or see how old the person’s name sounds (Agnes? Madison? Wilbur?).Basically, I am not impressed that someone is smart enough to trick a class full of 6th graders.

If you listen to education policy rhetoric, being interesting, contextual, and real-world are precisely the motivations for giving word problems.

Draw your own conclusions. :) :(