Approaching difficult problems and gaining insights

[JC2000]

I find that on encountering a tricky problem for which I have to look up the solution, I tend to 'understand' the solution yet over a period of time 'forget' it. On seeing the solution I understand the logical flow of the solution but on seeing the same problem over time I am often unable to recollect how to solve it.

With this I have two concerns :
1. If I was able to recollect the solution it would not really be beneficial as I am in effect memorising the solution and not really developing my capacity to solve problems. (Is this a fair assessment?)
2. If I try to break the problem down to the fundamental 'principles' applied or try to understand why that specific approach works over all others I feel overwhelmed by the 'data'. For every tricky problem involves a seemingly unique combination of fundamental principles or a unique perspective and thus keeping track of this for each and every tricky problem I encounter once again seems daunting and more of a memory challenge.

I believe that simply solving problems and struggling repeatedly over them will over time build a sort of intuition for the topic. To use an analogy, for a topic I have developed a feel/intuition for (say, Ratios and Proportions or Basic Arithmetic Operations), I somehow have the ability to navigate any problem with relative easy, simply because I spent a lot of time struggling with it in school (to the extent that I can apply these concepts almost unconsciously when they are required to used in some other topic entirely, for example the use of proportion in Stoichiometry). There could still be a whole collection of 'tricky' problems based on the simplest topics that I may still stumble over, but due to my existing 'intuition' for the topic I would be able to 'add' them to my 'inventory' much more easily.

But this leads to other questions :
3. When I learned 'basic' topics I was too young to really question how it was that I developed this intuition, I never consciously attempted to inventory approaches I encountered or basic principles to go by. Yet, I somehow feel that this approach of coupling 'blind faith' with solving lots and lots of problems is inefficient (Is it?).
4. So are there some 'Principles of problem solving' that need to be kept in mind while going through a new and unfamiliar topic (say Integration or Fluid Mechanics)?

 

[.Scott]

Looking back on any project that I have worked on, what I remember are the problems. Things that were easy to accomplish are easily forgotten.

But that works out just fine. I only need to remember the difficulties - since the next time I run into one of them, I will be better prepared.

So, to answer your questions as you posed them:
1) It is better to fully understand the solution, but it is beneficial to remember anything about it. Ultimately, many related examples will assist in creating an understanding.
2) Don't worry about remembering too many "tricky" solutions. In some cases, it may be decades before you run into the concept again, but you'll still have it.
3) You shouldn't have to consciously "inventory" approaches. Hopefully, you will remember enough about each concept for it to "ring a bell" when you run into it again. A young mind will look at a problem differently, but you always have a chance to look at the concepts again when you are older.
4) As a software engineer, I am constantly looking at new projects with new concepts. My only advice is to dive in with the full expectation that it is not going to be easy.

 

[JC2000]

Thank you for your response! I found the following bits of advice above most illuminating. Especially the last bit.

Things that were easy to accomplish are easily forgotten.

My only advice is to dive in with the full expectation that it is not going to be easy.

A young mind will look at a problem differently, but you always have a chance to look at the concepts again when you are older.

This is a perspective that I often overlook. When tackling a new topic or problem I tend to go about studying it as if it is the last time. Yet, even with the topics one is most at ease with there is almost always something left to learn (?).

Ultimately, many related examples will assist in creating an understanding.

Often a 'tricky' problem involves a principle/ approach which when used to solve more such problems becomes ingrained in ones repertoire.

The points you make (mentioned below) are often my biggest stumbling blocks.

1)...but it is beneficial to remember anything about it

3)You shouldn't have to consciously "inventory" approaches. Hopefully, you will remember enough about each concept for it to "ring a bell" when you run into it again.

My reason for asking for 'guidelines' or a checklist of some sort when tackling 'tricky' problems is what you mention in 1). When you say 'remember anything about it' what do you mean? Wouldn't it be some sort of inventory then?

Your point on about concepts 'ringing a bell' : is this ability honed solely through lots of practice (solving problems of varying difficulty and diversity but also problems of a similar nature)? Which is not an easy answer to stomach. Not because I don't want to solve lots of problems but because the approach relies on 'shooting in the dark' in some sense.

I also find that students who have more experience within problem solving tend to grasp concepts faster and have a deeper understanding much quicker (especially those that work towards Olympiads and so on from a young age). Given that many students begin solving 'Olympiad problems' at a very young age, they are unlikely to be making 'inventories' (?) but seem to develop a sort of 'muscle-memory' for learning concepts, and applying them to solve complex problems. I wonder what that means within the context of tackling new topics(?).

Thank you for your time!

 

[.Scott]

My reason for asking for 'guidelines' or a checklist of some sort when tackling 'tricky' problems is what you mention in 1). When you say 'remember anything about it' what do you mean? Wouldn't it be some sort of inventory then?

I suppose it can be a combination of deliberate and automatic. By "remembering anything about it": For example, you might remember that the sine function can be derived from a Taylor series without remembering exactly what that series is. But if you needed to know, you could reconstruct it.

Your point on about concepts 'ringing a bell' : is this ability honed solely through lots of practice (solving problems of varying difficulty and diversity but also problems of a similar nature)?

I can only speak from my experience. "Practice" sounds a bit tiring. Hopefully, you enjoy the work you are tackling - at least a little. So it will be "honed" or developed from experience - which will include your own attempts at tackling problems and interesting reading that you do about problems related to the kind of problem you can imagine you might some day encounter.[/quote]

Not because I don't want to solve lots of problems but because the approach relies on 'shooting in the dark' in some sense.

Indexing into the "inventory" occurs during "storage" and "retrieval". Rather than "shooting in the dark", continue to explore the problem - looking for more details and concepts. With some luck, you will find something familiar. Of course, you will often find something genuinely new to you. At which point you can look at the inventory of other folks (ie, research).

 

[symbolipoint]

Sometimes, similar or the general same problem is found often and can be generalized - just with different number values.

 

[Mark44]

I find that on encountering a tricky problem for which I have to look up the solution, I tend to 'understand' the solution yet over a period of time 'forget' it.

Things that were easy to accomplish are easily forgotten.

These two points go together, IMO, and are very important. It's easy to look at a solution and think "I get it," but since you aren't invested in the work involved in getting the solution, it's just as easily forgotten.

I think a better approach is to put time in on a problem first, and then if a solution is available, check your work against that solution, not the other way around, where the posted solution guides what you do in the problem. If they're different, go back to your work and see if there are flaws in your work or in assumptions you made.

 

[JC2000]

These two points go together, IMO, and are very important. It's easy to look at a solution and think "I get it," but since you aren't invested in the work involved in getting the solution, it's just as easily forgotten.

I think a better approach is to put time in on a problem first, and then if a solution is available, check your work against that solution, not the other way around, where the posted solution guides what you do in the problem. If they're different, go back to your work and see if there are flaws in your work or in assumptions you made.

Your point is a spot on description of what usually happens.

Often though (especially in Trigonometry or Integration) problems tend to have one manipulation which solves the whole thing. In that case, if the manipulation does not occur on the initial attempt then looking at the solution results in an inevitable 'I get it," moment which over time may once again be forgotten.

Of course your point is a valuable bit of advice! Thank you!

 

[JC2000]

Sometimes, similar or the general same problem is found often and can be generalized - just with different number values.

Yes this happens often, I tend to focus on the known and unknown variables!