I'm tutoring a very bright middle school student in a difficult geometry class. He understands most of the material, but struggles with some of the hardest problems and proofs. I find that when he goes through his homework, he gets stuck on the very hard problems and refuses to move on to another more than once, preventing him from finishing in time. I think his desire to solve every problem is good, but I am unwilling to give too much help with his homework, so in these problems I've given all the help I'm willing to give and he sits for hours trying to figure them out. I have tried to convince him he would be well-advised to move on and come back to the problem with a fresh mind, but he has a problem with skipping "too much" (more then one problem) for fear he won't get back to the skipped problems. How can I convince him to give the problem a rest for now and work on something else?
Nothing ironclad, but I have a couple of ideas. I hope they're of use.
It's admirable that he has persistence in solving problems. This is, of course, an important mathematical practice identified by many (not to mention codified in CCSS.MATH.PRACTICE.MP1: "Make sense of problems and persevere in solving them."
Most often teachers of mathematics must work to instill students with a sense that problem solving takes time, to counter a common view that "smart people" or "people who are good at math" are people who "get the answer," and quickly.
But persevering is only part of the story. It's important to help students develop an ability to learn with and from others (Martin, 2007). It's important to understand that collaboration is an appropriate activity when you're learning, and it's important to reflect on progress to know when one's own activity in solving a problem is productive.
Reflecting on progress (metacognition)
I know of no formulaic way to teach this, I only know that it can be helpful to get students to think not only about the problem, but about their own thinking about the problem. To take a slightly different example, there has been research asking students to track certain emotions during problem solving to gather data on their experience of problem solving (Lesh, 2010). Reflecting on this data helped some students realize that the feeling of discomfort that accompanied a difficult problem often came just before a necessary change in thinking. This changed the way they thought about (and experienced) those feelings in the future.
I'm not suggesting you turn reflection into a modelling activity (well, not necessarily), but one part of the approach taken in that study was a periodic approach to reflection. In other words, after a certain amount of time, the students were asked to record their emotions. In the case of OP's student, I suggest establishing a certain amount of time after which he or she reflects on whether the struggle is productive. You can probably come up with a good list of indicators of progress. Here are a few to start:
- What have we ruled out since the last check?
- What have we learned since the last check?
- Can we list one or more avenues we haven't ruled out yet?
In any case, with a decent list, you can help a student discover for him or herself just how productive the struggle is.
Learning appropriate collaboration is important (Martin, 2007). We rarely are completely alone in our investigations; our social resources include talking to other people, consulting books, searching the Internet, etc. Depending on the task, some of these are appropriate, and some are not.
However, it's almost always appropriate to at least talk about the problem with a peer who has not solved the problem yet (or a teacher/tutor who can listen and prompt appropriately rather than lecture). Compared to either struggling unproductively or looking up an answer, discussing the problem has distinct advantages. First, it can help a student expose thinking about the problem. Second, collaborating ideas means the possibility of having to argue and justify. Third, mathematical sense-making sometimes benefits from a change of perspective, which another set of eyes can help with.
These are skills that students often do not experience in a classroom, so if they can get experience engaging in them outside of the classroom, that is a kind of productivity beyond finding a solution. Though, of course, it can help get the student on a more productive path toward solution.
Lesh, R. (2010). Tools, researchable issues & conjectures for investigating what it means to understand statistics (or other topics) meaningfully. Journal of Mathematical Modelling and Application, 2(1), 16.
Martin, T. S. (Ed.). (2007). Mathematics teaching today (2nd ed.). Reston, VA: National Council of Teachers of Mathematics.
Consider pointing out how much each of the homework questions are worth. If his motivation is intrinsic, then how much will you learn from trying and failing to do this problem rather than doing the easier ones you already know how to do? If extrinsic, then note how much the problems are worth. Another thing I would point out is that if you do all the easier questions, you will have gained a better understanding of the material, and be better prepared to tackle the harder question. One of the most important things I've ever been told in mathematics is to "try some small numerical examples." It is annoying, but it improves intuition with proofs, and ultimately saves time.
Also, if one skips a bunch of problems, and doesn't have time to go back to them, then one wouldn't have had time to finish them all without skipping in the first place. Skipping doesn't cost as much time as it appears it will because often homework is designed to improve understanding.
Hope this helps.
In my experience (both as a student and as a tutor), the best way to deal with question on both tests and homework is a triage-style system:
- Look at all the questions, and take care of those that you know you can answer quickly; whatever you can't skip and come back to it ( = take the low-hanging fruit first).
- On the second pass, don't answer anything, but take note of (and maybe mark on the test/homework sheet) those answers that you known that you can answer for sure, but would take a little more effort (these are the ones that you should attack first on the third pass), and those that you're not certain that you can answer.
- Third pass: answer those questions you marked in the second pass.
- Then, if there's any time left over, you can deal with those monster questions that eat up so much time.
In my own experience as a high-school student, there is a possibility of a near-infinite loop between the third and fourth steps.....that's not a problem, as long as time is being spent productively, i.e. thinking about the questions as opposed to just looking at them.
One thing I hadn't even thought of until JPBurke mentioned it in a comment was that this method will help in a different way, namely that by looking at all the questions first, a student may be reminded of things that were known and forgotten, and has a great ability to help answer problems that were skipped the first or second time around.
I have a rule, which is called the 90% rule. Love it or hate it, it works for people in employment, as well as for students, and right now, this student, is not going to do well as a student, or in life unless they learn the 90% rule. Teaching them my 90% rule, will prepare them for exams.
The 90% rule, is that they need to aim to solve as much as they can within a fixed time frame. Its better to answer, and solve 90% of an exam, and miss out on 10% due to time constraints, then to focus on the 10%, and fail the exam miserably.
I would set fixed study times, 90mins max. At the end of 90mins, mark them on what they have solved so far. Tell them they have failed. After the 90mins, let them work on whatever, but everyday, dont help them for that 90minute period, and emphasize their mark, and if they failed or not and sure enough they will change their tactics.
In real life, I have seen people get fired for not-following the 90% rule. A process worker got canned, as accepted too many "difficult" repair jobs, following the 90% rule, he would have done the quick and easy ones, and after reaching minimum target levels could have done a few from the "too hard" basket. Fixing 10-20 complicated repair jobs when 60-80 they can do 60-80 repairs for the easy ones, makes them look slow, incompetent, and ultimately fired.
The value of people that can work on something until they solve it is worth almost zero, if you have nothing to show for an entires day work, and if anything, makes it clear to employers that you are not suitable, or smart enough for the job. In real world terms, doing 90% of your tasks quickly, and the remaining 10% in whatever time you have left, will let you keep your job.
Geometry can be very frustrating for bright students who excel with the linear nature of Algebra. That said, it is incumbent upon you to see what is stumping him. What is it about the "hard" problem that is causing him to be stumped about what to do next? Is there a pattern of difficulty? With Geometry, it's "what does what we know, tell us?" When he gets stuck, maybe it's time to list what we know and where each piece of the knowledge leads us. Also, working backwards can help. Perseverance is a great thing and I don't think we should encourage a student to give up. Certainly the problem has a solution and he should continue to work towards it. The problem here is not that he doesn't know when to quit but rather he doesn't know how to solve the problem.
While we've gotten good test-taking skills offered, I think they miss an important part of the question.
he has a problem with skipping "too much" (more then one problem) for fear he won't get back to the skipped problems. How can I convince him to give the problem a rest for now and work on something else?
1st. He has an (irrational) fear. We know he can skip the problem and come back but he can't do that. Even if he moves on to another problem, he'll still be thinking about it while he's trying to do the next. If he skips two problems then it only compounds the situation. You can not reason someone out of an irrational fear. They can only be convinced by experience that there is nothing to fear. Have him skip a problem on purpose (say #3) and then come back to it later. I like the idea of ranking the problems by difficulty but he's still going to run into the blockage problem eventually.
2nd. The value judgement that he needs to move on. The problem isn't that he won't move on, it's that he doesn't know what to do. Organizing his knowledge so he can see the connections between the different parts might help. I often had students in geometry who would get stumped because they didn't "see it". I would try to get them to break the problem down. What do we know? What do we want? These two lines are parallel. What are all the thing we know from that? These two angles are congruent. What are all the things that tells us? Maybe use index cards to record theorems and properties. Pull out the ones that apply to that problem. You mentioned proofs. Can he do a flow-chart or paragraph proof or does it have to be two-columns?
Sorry, but it just frosts me that as educators we would encourage him to move on.