I propose that “being stuck” is a terrible way to spend one’s time learning.
Making the task unnecessarily hard for oneself is simply a waste of time.
There is a difference between “research problems”, and homework exercises.
Research involves exploring uncharted territory involving genuinely hard things for which no one can see a clear path to a solution. But being stuck with no clear way to solve the class of problems called “homework exercises” (or at least a way to solve the meta-problem), results in lots of wasted time non-systematically trying all kinds of things until something sticks. Doing such a thing does not make anyone become a better problem solver.
“Being stuck and not knowing what to do” should not be the “normal state of affairs” for homework exercises. A well-designed curriculum should have a smooth, effective learning curve / experience. Prerequisites should be made clear so that a learner, when “stuck”, can quickly identify their knowledge gap and plug it before proceeding.
There is a standard meta-strategy for tackling problems:
Do I have methods I can try which are likely to succeed? If not (i.e. “stuck”), then first solve the meta-problem of finding such methods.
Then our new problem is: do I have high-success-likelihood meta-methods to find the above-mentioned methods? If not (i.e. “meta-stuck”), then first solve the meta-meta-problem of finding a meta-method to find such methods.
In a classroom setting, often the most productive thing to do is obvious: ask your friends (or teacher) to give you some guidance (point you in the right direction, but not provide the answer).
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Sometimes maybe the student lacks meta-cognition to tell if they are “good stuck” or “bad stuck”. If there are people available to offer guidance, it’s good to just ask.
If I’m a learner, and I find myself in a situation where I am not engaging in learning activity I know to have high likelihood to grant me progress towards my learning goal, then I am by definition “meta-bad stuck”, because I cannot tell if the method I’m employing is productive or not.
If I don’t know whether what I’m doing is productive or not, then chances are I’m not being productive.
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For solving homework exercises, there should always be a clear route, or at least meta-route:
The necessary (prerequisite) concepts are supposed to be already covered. Just put them together in different permutations. Even if you don’t solve the exercise, going through these permutations gives you real mastery of the concepts.
After you’ve tried all permutations you can think of, and still cannot solve the problem, there are three main possibilities:
- The learning material sux and the exercise requires knowledge of stuff you have not learnt yet.
- You have not properly mastered the prerequisite concepts.
- It is your natural blind spot; by definition it is very difficult to identify it.
If you’ve been diligently self-testing / revising, then (2) is unlikely, and you can just rule it out.
Maybe you have some meta-knowledge which tells you that it’s (1), then just skip the exercise and return when you’re ready.
Maybe it’s (3), or might be (1) but you’re not sure.
Then just ask a friend / teacher. Or consult other learning material.
It is a terrible waste of time to continue to agonise over a problem hoping for a miracle to happen, long after you’ve already gone through all the reasonable permutations.
If later you solve it magically, it’s probably the magic of your unconscious mind fitting things together behind the scenes. In which case there was no point agonising over it. The unconscious mental processes could have done its work without your conscious interference. And you could’ve been working on something else instead. (Cf. “diffuse mode”, “chunking”, etc. https://www.coursera.org/learn/learning-how-to-learn)
TL;DR: “bad stuck” = unknown unknowns; “good stuck” = known unknowns.
Not all effort is created equal.
The sort of effort we should be praising is effort that leads to growth. Where effort does not result in growth, we need to direct students towards the type of effort that does.
Guessing is “hard work”, but it’s a bad learning strategy. (Especially not for mathematical subjects in which rigorous, water-tight deductive reasoning is necessary.)
Guessing might work in the short run. But not particularly viable for mathematical stuff relying on many concepts combined together. It just makes learning the later stuff more (unnecessarily) difficult.
Did you guess?
One aspect of assessment I stress to my students is to not fool themselves because they scored 18 correct out of 20…especially if they guessed on 5 of the questions. This student really only knew 13 of the 20 answers. What do students (and teachers) do, though? We see 18 out of 20 and think little Maggie knows 90% of the material. But, that is not true.
New research argues that optimal learning is achieved when we face a task at which we have an 85% accuracy rate. Training at this level dramatically improves the rate of learning. The bad news is that this magic formula is calculated on the basis of training an artificial-intelligence algorithm. However, there are grounds to believe it applies to our brains as well.
Making learning unnecessarily difficult doesn’t help anyone. And helping students breeze over content so they’ll get a good grade without learning doesn’t help anyone.
“engagement alone, however, is not sufficient for learning, and although learning is hard work for students, not all hard work leads to learning.”
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If students are taught how to learn early in the curriculum, they will be more likely to excel throughout their academic careers, as practitioners, and as life-long learners.
We ask our students to teach patients and other practitioners, now and in the future. We ask students to teach each other through in-class presentations or cooperative learning strategies. We ask students to serve as peer-tutors or teaching assistants. However, as we focus on students’ ability to apply, evaluate, create, or communicate information, we may need to teach students how they learn so they can be more effective learners — and ultimately teachers — themselves.
