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A moment ago, I happened upon https://puzzling.stackexchange.com/questions/99712/almost-impossible-sudoku-like-puzzle which explicitly mentioned that this student was given a puzzle in math class that his math teachers couldn't solve. And it got me thinking: To me, it feels like a teacher like that is unfit for teaching. After all, how could they impart knowledge they don't have onto their students?

Should a teacher be able to solve all the assignments they give their students themselves? Assuming the assignment is in fact solvable obviously.

Note: The question that prompted this was a math problem, but I'm looking for course-agnostic answers, if possible.

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    From the linked question, it is not clear to me that the puzzle was actually part of the course (instead of: a little fun for good students with no real relevance to grading and the course topic) and if the teachers who gave the puzzle are the same who couldn't solve it. – user111388 Jul 7 at 13:21
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    From the George Dantzig article in wikipedia, “In statistics, Dantzig solved two open problems in statistical theory, which he had mistaken for homework after arriving late to a lecture by Jerzy Neyman.” A delightful and famous exception! – Ed V Jul 7 at 13:47
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    For required work, yes, but for extra credit, no. – John Coleman Jul 7 at 23:20
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    In "How To Solve It", George Pólya writes: ""There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann didn't say anything but after five minutes he raised his hand. When I called on him he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann." [2nd ed. (1957), p. xv]" – Count Iblis Jul 8 at 12:01
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    @EdV While Dantzig's story is certainly great, his professor didn't assign those problems to anyone. It's common in upper level courses to point out some of the major open problems in the field. – Kimball Jul 8 at 15:17

16 Answers 16

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In general, yes, a teacher should know how to do any assignment and, in some cases, should have actually done it. In teaching programming, for example, it is usually a mistake to assign a problem that the instructor hasn't essentially done themselves.

The reason is that one of the tasks in making assignments is to estimate the effort and time required to do the task. If you can't solve the assignment yourself, you are giving an open ended task to students. In almost all situations students have a limited time to spend on any given assignment and that competes with their other tasks.

There is an exception to the above. If you tell the students at the beginning that you are assigning something you don't have an answer to, and will be grading their efforts rather than their results, then you can make this work. The assignment becomes an exploration. At lower levels of education this makes less sense than it does at higher (say, doctoral) where it is natural to explore the unknown.

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    At school I used to hate doing problems (or science experiments) that had already been done. It seemed to me that they weren't actually problems or experiments any more. Pupils are motivated in different ways. I personally would have worked ten times as hard if I'd known the teacher didn't have an answer. As it was I often skipped the set work and did my own investigations (blew things up once or twice!) – chasly - supports Monica Jul 7 at 20:43
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    @chaslyfromUK, I do completely commiserate with you, but, on the other hand, why would/should we think that the next gen of kids can so easily become wiser than the previous? E.g., if all that "school knowledge" is really so useless, ... ??? – paul garrett Jul 7 at 23:09
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    @paulgarrett because there's a lot more data now than there was before, and there will be exponentially more data in the future. Combine that with the exploratory assignments Buffy ends on and it's a lot easier to assign (and execute) assignments exploring the unknown. I've even seen undergraduate courses on data mining that would get a lot of new real world data each year (or even between semesters). Getting big chunks of data and a large set of tools makes it more fun for everyone. And sometimes it leads to actionable insights that can be fed back to where the data came from. – JJJ Jul 8 at 0:11
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    @chaslyfromUK there's some decent evidence from physics lab pedagogy that you're right, and students get almost nothing from just following a set of instructions to do an experiment (other than familiarity with lab equipment). What works much better is giving a goal and equipment and letting the students figure it out, but this is much harder to organise and grade – llama Jul 8 at 5:21
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    I tell my students if they cannot show that they can reproduce a known experiment, no-one will believe them when they show any new results. Even in real scientific experiments, you usually reproduce a known set of results first, to ensure your equipment is working properly. – Marianne013 Jul 8 at 9:07
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In general the teacher must be capable of achieving what is they want their students to achieve by undertaking the assignment, but that might not be coming up with a solution.

In the modern world where all knowledge is at our finger tips all the time, the job of an educator is much less to impart knowledge, and much more to guide and mentor students, help them learn where to find information and assess its reliability and to concentrate on the higher level cognitive skills, such as problem solving, synthesis and reflection.

This means that often coming up with the answer isn't the point of an assignment, but rather something about the journey to getting to the answer (or failing to do so).

Taking your example: It could be about taking a set of puzzles and working out what the common rules are to distinguish solvable from unsolvable problems. Or deducing if the difference between hard and easy puzzles is quantitative or qualitative. It could simply be about learning that some problems are not soluble, but there is still value in working on them.

Many of the other answer here suggest this kind of approach is only applicable at higher levels, like graduate school, but the editor in chief of the AMS' maths education blogs talks here about giving unsolved math problems as homework to undergrads, and Lior Pachter talk here about ones that you could give to K-12 students.

My own maths education started incorporating this sort of "Investigation-led" learning at 15 as part of the UK national curriculum. While the problems set were not insoluble (how many bricks do you need to build pyramids of a height n, and deriving the basic rules of differentiating polynomials empirically), they shared in common that the journey not the end point was the purpose.

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In engineering, when students are asked to solve real world open ended problems,sometimes it works out that there is no solution. Figuring that out is an important outcome.

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    In the case where there is no solution, why shouldn't the teacher also know this fact? – user111388 Jul 7 at 13:19
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    @user111388 because some problems do have solutions but the solutions have not been found yet. – Solar Mike Jul 7 at 13:35
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    @SolarMike: Sorry, I do not understand. Why is it not okay that a teacher doesn't know "the answer is X" when the answer is X but it is okay that the teacher doesn't know "there is no solution" when there is no solution? Or am I misinterpreting the answer? – user111388 Jul 7 at 13:39
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    @user111388 Because conclusively proving that there is no solution is often itself not possible, or at least has not been done. Wikipedia has a nice page of things believed, but not known, to be true in mathematics. – Tiercelet Jul 7 at 21:14
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    @Tiercelet: I still don't understand. It is not okay if a teacher can not solve a solvable assignment, but if the assignment is unsolvable, it is okay if the teacher believes it has a solution? Also, I don't quite understand the relevance of the link - those problems are research problems, (hopefully) not graded assignment problems! – user111388 Jul 8 at 5:41
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I remember a friend of mine reporting from an oral final exam in graph theory by a professor with a certain renown in the field. After a number of questions he was able to deal with gracefully, the professor asked him to prove some theorem. He dragged out basically the complete toolbox and made a number of attempts but each time wasn't quite able to close the final gap. Finally the professor aborted his tries and told him "it's ok, we managed to prove this one only last month".

The grade was the best. Basically the professor checked at what level and with what aim the student floundered. Which tells more about the actual problem-solving skills of a student than the ability to reproduce a preexisting proof.

Which doesn't mean that such an exam is pleasant to be in.

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My short answer to the question is: yes.

Long answer is as follows:

Professors/ teachers should definitely have the knowledge that must be passed onto the students. I find it hard to think of an instructor who attempts to teach a topic that they themselves do not understand. However, again, you must be aware that there are different levels to understanding and even a professor might not understand particular topics very deeply (everyone is human with their own strengths).

So in general, for levels under graduate school, I believe that the instructor must be able to solve the assignments they give to their students.

However in grad school, the nature of assignments change. Of course there are still homework in most doctorate and masters classes but a professor is also directing research of their graduate students and sometimes might assign tasks that they might not be able to do themselves or it is uncertain whether they could do it (a yet unsolved problem would belong to this category).

So, in summary, for education that encompasses teaching of certain textbook knowledge, I believe that the professors/teachers must be able to solve the assignments they give to their students themselves. But for graduate school and especially for research, this requirement breaks down.

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I have always felt obligated to solve every problem out myself before handing it to a student. I put myself in the student's shoes to see if an assignment is of good quality. This comes at a price in the way lessons could be developed further, I could cover something else, etc.

However, this teacher is ABSOLUTELY NOT unfit to teach! I have never been very good at puzzles and fancy tricks. It does not make me bad if I am not able to solve it.

My mentor told me this, and I think this comes with age and wisdom. It is important to choose topics you feel are worthwhile to learn at the price of refraining to learn other subjects. Learning tricks like these falls into this category for a lot of teachers.

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Giving students an assignment which a lecturer can't solve generally should not happen. Sometimes it can be justified (e.g. students involved in a real-life research project, where a problem can have many solutions or none at all), but it has to be made clear to students.

However, one mistake does not make anyone unfit for teaching. Just as students, lecturers need time and process to learn how to teach and be better in their roles. Doing mistakes along your learning pathway is completely normal, but of course one should reflect and learn from them.

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If assignments are about "knowledge," it might seem that a teacher who does not know the answer to the assignments is unfit to teach. Knowledge, however, is not the end of many (most?) lessons. In many cases, assignments are about skills more than knowledge, and a teacher does not necessarily need to be able to complete all the assignments themself to be able to teach the skill to a student. A basketball coach does not need to be an excellent basketball player themself to teach players how to excel at basketball. Being excellent at teaching a skill is a different, and sometimes non-overlapping, quality from being excellent at the skill itself.

If assignments are about skills and processes, the teacher knowing the answer ahead of time, or even being able to complete the assignment themself, is not necessary for the teacher to be excellent at teaching students the necessary skill. It is sometimes possible to teach skills (and even do it very well) that one does not personally have.

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I don't think, particularly for extension problems, that it is vital the teacher has solved the problem unaided. However it is certainly desirable, just because they will have a better idea of the difficulty if they have done so.

However, what is important IMO is that the teacher has seen and verified the solution, since otherwise how can they be sure that the problem can be solved? For the linked problem it was by no means obvious to me that there was a valid way to arrange the numbers. (Of course you could ask for either a solution or a proof that no solution exists, but the difficulty of the problem will then be very different depending on which of these you have to do, so the teacher really should know which is the case.)

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    I guess I disagree. When a student comes to you for help on a problem for which you have not done the solution, how do you guide them? It is very hard to give them the "minimal hint" that will get them over a rough spot and help them achieve insight. Very hard to know where they've got wrong and need to be put right. – Buffy Jul 8 at 16:16
  • @Buffy Without looking at a maths problem, I can always recommend trying to start from the other side of the problem (i.e. pretend it's finished and work backwards) or "if it's impossible, can you prove this / think of an argument for why this is?" or "can you use symmetry to help you?". These 3 ideas apply to every maths problem ever but the last one may well help in this instance, depending on the level of the student. So yes, a teacher, an expert in their field, can help a student without knowing the question, nevermind the solution. – stuart stevenson Jul 9 at 22:35
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People say that you only understand something properly when you can explain it to others. So, if we say that the teacher has set the students to complete a Sudoku logic puzzle, then as long as the teacher understands and can clearly explain the work, it is no big deal if they can't solve the Sudoku themselves.

However, it is doubtless infuriating when a teacher cannot do the Sudoku, cannot explain how a Sudoku works and does not attempt to complete the Sudoku, yet gives it to their students. As long as the teacher can offer help to the students then they are doing fine.

In my experience, teachers generally know what they are talking about/teaching. However, there are some cases when a teacher has not had much idea about what it is they are instructing on, but I have rarely known a teacher to not attempt/explain the task itself.

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  • Actually, Sudoku might be a poor example. Some are much harder than others. Being able to solve it "in theory" might not be enough. – Buffy Jul 7 at 20:44
  • Good point. I used Sudoku because it is an example of knowing how to do something rather than actually knowing the answer, but perhaps there are more apt analogies. – monsieuralfonse64 Jul 7 at 20:57
  • @Buffy: To me, from a mathematical perspective being able to solve a sudoku in theory (via recursion) is enough. There is no mathematical need to know any of the tricks that fast sudoku solvers know. – user21820 Jul 10 at 9:26
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Should a teacher be able to solve all the assignments they give their students themselves?

I'd say yes provided that we are talking about a non research level, where the assignments are intended to prepare students for exams. In this context, if the teacher cannot solve a problem given to the students, then the teacher probably is not qualified and/or prepared to the class and thus should not be teaching that subject.

As advised by Krantz in his book How to Teach Mathematics:

If you are going to stand up in front of thirty people or three hundred people and try to teach them something, then you had better

  • Believe that you are well qualified to do so.
  • Want to do so.
  • Be prepared to do so.
  • Make sure that these characteristics are evident to your audience.
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  • "Believe that you are well qualified ..."? No... You had better be well qualified to do so. There are many cranks who believe they know better than professional mathematicians, want to teach others their crankery, and even prepare lots of 'material' to do so, and definitely these characteristics are evident to anyone with rudimentary mathematical training. – user21820 Jul 10 at 9:24
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Have you considered that maybe the teacher did know how to solve the problem (it is an elementary problem) but was using the white lie that they couldn't solve it for motivation? When I was a teacher, I would do things to "model the behaviour" of going from not knowing to knowing and sometimes that means pretending that you don't know the answer when of course you do.

Also, on the topic of puzzles, some things are better solved by a large group than a single teacher, for instance problems requiring a high degree of computation or (in the case of the 8 queens problem), combination.

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The sort answer is yes if the students learn something and they appreciate the experience.

Knowledge Transfer Perspective: When a teacher presents a good open problem to students, she offers valuable information. A good open problem is one which a lot of people are interested in and for which no known solution exists. If, in addition, the teacher herself has attempted to solve the problem, then she can present to students the approaches she tried, showing why they failed. This adds even more to the value of a good open problem. In the case of a "bad open problem", when not a lot of people are interested in it and/or an actual solution does exist unbeknownst to the teacher, the instructor betrays her incompetence.

Teaching as Service Perspective: A lot depends on the rapport of teacher and students. If no student in class can solve an assigned problem, then students might view, often justifiably, their work as a waste of time. If the problem was chosen correctly so that students are challenged and are successful at solving it, then everyone feels they achieved something and knowledge retention is, likely, improved. Assigning an "impossible" problem (whether the teacher can or cannot solve it) is just putting students down and failing to get students excited about the subject matter. This is a signal of another form of incompetence, that the teacher does not understand her students' needs.

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I'm in my undergrad and have asked myself this question in the past. I think a teacher should be capable of scoring in the top 10% of students in the same conditions (time, amount of notes, etc.), and near max points if no constraints.

There is no doubt a gray area when the class is very interdisciplinary.

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In general, yes teachers should and must know the answer.

But in very advanced classes such as some specific PhD or Master's classes, the teacher can challenge the students to solve an unsolved problem, or at least explain why a problem is unsolved.

One example is the traveling salesman's algorithm which doesn't have an optimal solution. (please correct me if I am wrong)

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  • I should have clarified from the start that I'm referring to solvable assignments. – Nzall Jul 7 at 13:35
  • "optimal" can be most profit, least cost, minimum time or minimum distance... – Solar Mike Jul 7 at 13:36
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    Why shouldn't a teacher challenge the students in lower classes? – user111388 Jul 7 at 16:04
  • @user111388: I don't see anywhere were Herman Toothrot (or a commenter) said anything about lower classes (challenging them or not challenging them). – Dave L Renfro Jul 7 at 17:47
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    Yes, Traveling Salesman has an optimal solution since it is a finite problem. The issue is that there is no sufficiently efficient general algorithm for finding it. But the solution exits. See: en.wikipedia.org/wiki/Travelling_salesman_problem – Buffy Jul 7 at 23:43
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George Dantzig was mathmetician who famously solved two "homework problems" set by his lecturer, that were in fact unsolved problems in statistics. So if you happen to be teaching a genius then yes, it's a great idea to set problems you can't solve yourself!

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    So, the story says the professor DID say they were unsolved problems, but Dantzig was late to class that day, and did not hear that part. – GEdgar Jul 9 at 20:27

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