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When I was a student, it was very much frowned upon to ask for help without making an effort, like how math.stackexchange.com operates (for the most part).

In the high school where I work, it is common for teachers to go through full solutions with students who want help despite having made no effort themselves. I know that for questions that are very simple, it takes little effort to go through a full solutions with a student. But for a question like "For which values of $m$ is the line $y=mx+6$ tangent to $x^2+y^2=9$?", I do not feel like helping students who claim that they have no idea where to start.

Is there much value in going through full solutions with students? Or is it better to dismiss students by giving them a hint and telling them to go away and attempt it seriously?

Related question: I also often have students asking me to check their work, expecting me to find an arithmetic slip or sign error they have made, when for example they are computing a derivative or integral. Can anyone advise how they handle these sorts of questions? Tell them to check it on Wolfram-Alpha?

I hope this is not too opinion-based.

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    $\begingroup$ What was the knowledge level? The more jargon, the more implicit equations you bring up, the more stuff you are going to have to remind them of. If you start off the class with circle equation, the idea of tangents and the idea of slope intercept form given, of course they aren't going to have any idea where to start if any of those ideas are new or very old. You'll need to remind them what tangent even means, what the equation of a circle looks like, and if its new to them, what SI form is. And if this stuff is very old you'll need to do the same thing, even if you're in a calc class. $\endgroup$ – whn Mar 6 '18 at 15:20
  • $\begingroup$ What is MSE? Please edit and type it out $\endgroup$ – user985366 Mar 7 '18 at 23:49
  • $\begingroup$ @user985366 Math Stack Exchange $\endgroup$ – A. Goodier Mar 8 '18 at 18:15
  • $\begingroup$ thx. my best guess was "mean squared error" $\endgroup$ – user985366 Mar 9 '18 at 14:49
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    $\begingroup$ I do not feel like helping students who claim that they have no idea where to start. Why? You're most likely dealing with a person who has at least 2 out of 3 of the following issues: (1) had underqualified K-12 teachers, (2) isn't highly intelligent, (3) has only experienced math as a set of algorithms to be mastered. If they say they have no idea where to start, they're probably telling the truth. That doesn't mean you have to solve the problem for them. Draw them out. Ask them if they can define terms like "tangent." Ask them if they know what shape is represented by $x^2+y^2=9$. $\endgroup$ – Ben Crowell Mar 12 '18 at 2:08
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My advice is to use Socratic dialogue and lead them through the process. Get some kind of interaction, engagement.

It is a balance. You can go all the way to a boot camp sergeant sending the maggot away to someone who shows the whole solution. Some students who are used to more explanation may find Socratic dialogue uncomfortable and resist but even then if you explain that you are trying to get them involved, it is good enough. [If they still won't engage, let them sink. I mean it. Life will be harder than school for sure.]

If you do a Socratic explanation or even a full explanation MAKE SURE to give a follow-on problem (or five!) so that they can drill the learning, prove they can use it, etc. Humans are not computers. Just showing us a method is not good enough so that we can use it. We need to practice it.

P.s. It is absolutely in zone as a valid question. Deep in the pedagogy center.

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    $\begingroup$ "Humans are not computers. Just showing us a method is not good enough so that we can use it. We need to practice it." Actually, if you ever try to deploy untested code to Production, you will learn that computers need to practice, also. ;) $\endgroup$ – Wildcard Mar 6 '18 at 4:46
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    $\begingroup$ Or as Donald Knuth put it: "Beware of bugs in the above code; I have only proved it correct, not tried it." $\endgroup$ – Wildcard Mar 6 '18 at 4:47
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    $\begingroup$ @Wildcard: You mean programmers need more practice. The computers don't need any practice; they are just doing whatever they're told. =) $\endgroup$ – user21820 Mar 6 '18 at 10:28
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    $\begingroup$ I do the same with children, you have to be the voice that should be in their head. $\endgroup$ – the_lotus Mar 6 '18 at 15:41
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I'm going to add in here as the mother of one those kind of students, and say in addition to the advice you've already gotten, also consider the student and what actual difficulties they may have (whether they have an IEP, 504, or not) and if it will help them or improve your relationship with them.

My kid has times when she truly is befuddled with these kind of things (not just in math), and she has to already trust the teacher just to even ask for help. In math it might take a couple times for her to get it, but once she gets it, she gets it. Thoroughly and completely. She's in Algebra II now and is tutoring a kid taking Algebra I (this is a huge social advance for her). If she had not had teachers that gained her trust and were willing to work with her, I can guarantee you this would not have happened.

Having teachers she trusts that are willing to work with her makes a huge difference in her overall effort in a class, and her grades prove it in their ups and downs. If she doesn't trust enough to ask for help, and get that help, she gives up. Completely. Doesn't matter what the subject is. She thinks she's stupid because her classmates don't understand her jokes, but the majority of people who get her jokes have a master's degree or better. She just has issues she was born with, and any decision you make to help (or not help) a student can ripple out for years and affect many lives.

Every student is different - learns differently, needs different motivations, different help. I would ask you to not pick a blanket way to approach this issue, but to consider each student and their needs and strengths before making a decision.

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    $\begingroup$ IEP? 504? This is an international site, so it's helpful to explain (or avoid!) local jargon. $\endgroup$ – Peter Taylor Mar 6 '18 at 8:49
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    $\begingroup$ @PeterTaylor IEP stands for Individualized Education Plan and a 504 is a "plan developed to ensure that a child who has a disability identified under the law and is attending an elementary or secondary educational institution receives accommodations that will ensure their academic success and access to the learning environment." I'm not sure what term to use to describe this group of students. I think the point is, OP should consider the possibility that the student really doesn't know how to get started, through no fault of their own. $\endgroup$ – stannius Mar 7 '18 at 0:33
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In most cases just giving the solution is easy but not fruitful. Student learn by their own work. But many student can be honestly stuck, not in a position to do anything if one simply tell them to "really try".

My experience is that student who do not know how to start an exercise actually do not know what "really trying" means. They do not need a solution (which they would more often than not try to learn by heart without understanding it), but a method. And this needs some thinking (at least, it took me years to get to answers I find approximately satisfying)

For example, today I was asked how to prove that, given three subspaces $A,B,C$ of a vector space $E$, the following formula holds true: $$ (A\cap B)+ (A\cap C) \subset A\cap (B+C) $$ I went through the following points:

  • realize that the central symbol of the formula is $\subset$: we want to prove an inclusion (this is not obvious to all students!),
  • one should check in one's lecture notes or book the definition of $+$ if the meaning is not obvious (sums of sets should trigger a warning, but of course the lecture has defined this for subspaces); although they had more than a week to prepare this exercise, only one seemed to know the definition of $B+C$,
  • one should remember how an inclusion is usually proved, e.g. by looking into previous exercises of the sort,
  • then one can actually solve the exercise: one considers an arbitrary element $\vec{u}$ in the left-hand-side of the inclusion, and show that it belongs to the right-hand-side; then use the definition of $+$ to exhibit a decomposition of $\vec{u}$, etc. You got the idea.

The actual solution is the last point, one fourth of my answer. In retrospect, I should probably have left it out, and reassign the exercise for the next session to avoid too many students focusing on it and forgetting the first three points, which are actually much broader in scope and much more important to their studies.

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    $\begingroup$ If you use that set-theoretic notation and some students don't even know what it means, then it is simply a pedagogical failure even before you reach this question... Too few people realize that the vast majority of conceptual problems in mathematics arise from inability to grasp basic logic and reasoning. It is really a mistake to think students can learn or pick up logic along the way. Only a small fraction can do that on their own, and the rest will just guess their way through each course without ever having 100% understanding of anything. $\endgroup$ – user21820 Mar 6 '18 at 10:33
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A lot of it depends on context. However, one thing that may help both sides (you with your time and them with their problems) out is to, instead of working the problem for them, make them work it in front of you. You can prod them with questions until they see how it works. Sometimes you have to get very close to just giving them the answer, but, since they have to think about it themselves, they won't go to you as a replacement for thinking about it themselves. Make them fumble through the book looking for an answer. Just keep asking questions until they figure it out.

If they are truly having trouble, this will equip them with the tools they need to succeed. If they are trying to skip the "hard work" step, this can actually require more time and work from them, as they have to jump through every question you ask, right there in front of you.

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Your students are in high school, which means that there are limits to how much responsibility you can expect them to take for their learning. And you're already most of the way through this year, so it will be difficult to apply this to this year's class. So what follows is the extreme version, and you'll have to decide how strictly to apply it, and you'll probably have to wait until next year with a new class to establish that This Is How It Is.

You should teach your students how to do problems. When your students need to do a problem, your lessons should show them how to do it. If they are having trouble applying your lessons to the problem, they should be able to show what parts of your lesson they have applied, what part they are having trouble applying, and why they are having trouble.

If a student wants to ask you for help on a problem, they should have a list of key concepts for the problem, lecture notes from when you went over those concepts, what work they've done applying those notes, and questions about how to apply them. If they don't have that ready, walk away.

As I said, this is an extreme version, and you'll probably want to not go quite this far, especially at the beginning. But you should be asking them to show effort on their own, and refuse to help if they aren't putting any effort forth. Ask leading/Socratic questions, and as the school year goes on, ask fewer, less leading questions, and demand the students do more and more on their own (not necessarily more and more of the actual math work, but more and more of the meta-learning stuff: identifying what the key concepts in a question are, looking in the book/lecture notes/internet/etc. for those concepts, writing down lists of "things I'm told" and "things I'm supposed to find", identifying applicable formalae/algorithms/etc.)

In education, there is the metaphor of "scaffolding": if you're building a building, you may need a bunch of outside supports as you're building, but if the building still needs those supports at the end, you haven't done your job. Your job as a teacher is to take away support, slowly enough that your students aren't left adrift, but fast enough that by the end they can do the work on their own. Getting the balance right is one of the major challenges of teaching.

If they want to check their work, an option is to ask another student (assuming you don't have concerns of students copying each other's work). Other options would be giving them the answer, or, if you're worried about them just taking the answer and not doing the work, giving them a list of answers (that is, the answers from all the problems, plus several other decoy answers, with the order scrambled) so they can check whether the answer they got is on the list, or giving them a "hash" of the answer (some property that is unlikely to be shared with an incorrect answer, but is not enough to get the answer). But if this is homework, they shouldn't be stressing out on getting the right answer in the first place. You shouldn't be taking away a significant points for getting homework answers wrong, especially if they are easy to miss errors (if they get an answer that's obviously wrong and don't notice, maybe that deserves some points taken off). The whole point of homework is to have a chance to make mistakes before the test.

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I'd understand if this was at a higher level, but we're talking high school here. You are expected to equip them with at least the bare minimum of knowledge needed to pass, because the vast majority are not going to care much about pure mathematics past HS but the topic is sadly mandatory for an unnecessarily long period of time in most places in the world.

That in itself makes going through full solutions worthwhile - at the very least, this gives those students who don't know where to start (the very same you pointlessly and erroneously deride, as, again, you severely overestimate the importance of your topic for most students) a full guideline that they can then study for the one and only time when knowing it actually matters. Until this changes and you only have to deal with students who are there primarily for maths, no other solution is acceptable, and you're only going to find heavily biased solutions here that do not fit the context you are in.

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    $\begingroup$ Whatever the level, one should aim for actual understanding, not shallow memorization (this implies the level of a course must be adapted to the context, sure). $\endgroup$ – Benoît Kloeckner Mar 7 '18 at 14:03
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Just personal experience, don't know if it's a recognized method.

When my father used to help me, he would show me the full working out and answer on a blackboard. When he was satisfied that I understood how he obtained the answer, he would erase it all, and then thee me it was my turn to work it out.

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Why don't you:

  1. Give a hint
  2. Go through the full solution if they don't get the hint
  3. Let them do another similar problem in front of you to confirm their understanding

?

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