Problem solving and so frustrating

[Bliman]

Hi all,
I am learning by myself calculus and I am on page 865 of the Larson book of Calculus. I also watch Professor Leonard for explanation.
Till now everything is going relatively smoothly. Besides one thing and that is problem-solving questions.
When I get to those everything falls apart and it is getting really frustrating. I was speaking to my doctor about the math I was doing. And she said something I can still recall. Everyone can plug in numbers and do the number crunching and learn formulas but not everyone can think.
This is how I feel. I can do the exercises ok, but when the problem solving comes I crumble. And it is so frustrating. It seems like my brain gets garbled when I look at these and my mind becomes blank. Even when I try to get as much information out of the problem.
The problem is that I don't know how I can get better at it. Or if there are books how I can learn about that.
Take for example this problem.
"
A bomber is flying horizontally at an altitude of 3200 feet with a velocity of 400 feet per second when it releases a bomb. A projectile is launched 5 seconds later from a cannon at a site facing the bomber and 5000 feet from the point that was directly beneath the bomber when the bomb was released. The projectile is to intercept the bomb at an altitude of 1600 feet. Determine the required initial speed and angle of inclination of the projectile? (ignore air resistance).
"
Now after looking it up on the internet I finally got it when the answer was shown (although I still don't get why you would choose the angle to be 0 when the bomb is released).
How do you people know that the angle is zero when the bomb is released?
The point I am trying to make is that I have great great difficulty with all these problem-solving questions. Sometimes they seem to get things out of thin air. From somewhere I have never thought of. I have so many different problems with it. And I want to get better at it.
Can someone tell me what books are good for that? Or what I must do? Am I not smart enough? It is so frustrating.
It seems like you need another mindset to solve these problems.
So I can use all suggestions I can get to get better at it.
Thank you

 

[lev888]

Hi all,
I am learning by myself calculus and I am on page 865 of the Larson book of Calculus. I also watch Professor Leonard for explanation.
Till now everything is going relatively smoothly. Besides one thing and that is problem-solving questions.
When I get to those everything falls apart and it is getting really frustrating. I was speaking to my doctor about the math I was doing. And she said something I can still recall. Everyone can plug in numbers and do the number crunching and learn formulas but not everyone can think.
This is how I feel. I can do the exercises ok, but when the problem solving comes I crumble. And it is so frustrating. It seems like my brain gets garbled when I look at these and my mind becomes blank. Even when I try to get as much information out of the problem.
The problem is that I don't know how I can get better at it. Or if there are books how I can learn about that.
Take for example this problem.
"
A bomber is flying horizontally at an altitude of 3200 feet with a velocity of 400 feet per second when it releases a bomb. A projectile is launched 5 seconds later from a cannon at a site facing the bomber and 5000 feet from the point that was directly beneath the bomber when the bomb was released. The projectile is to intercept the bomb at an altitude of 1600 feet. Determine the required initial speed and angle of inclination of the projectile? (ignore air resistance).
"
Now after looking it up on the internet I finally got it when the answer was shown (although I still don't get why you would choose the angle to be 0 when the bomb is released).
How do you people know that the angle is zero when the bomb is released?
The point I am trying to make is that I have great great difficulty with all these problem-solving questions. Sometimes they seem to get things out of thin air. From somewhere I have never thought of. I have so many different problems with it. And I want to get better at it.
Can someone tell me what books are good for that? Or what I must do? Am I not smart enough? It is so frustrating.
It seems like you need another mindset to solve these problems.
So I can use all suggestions I can get to get better at it.
Thank you

Please post the solution you found - it's not clear which angle you are referring to. The initial angle of the bomb?
Regarding problem solving - it's a skill that you can improve. There are helpful steps that apply to any problem (read it carefully, visualize what's happening by making a diagram, etc.) and steps that apply to the specific area of math. If the problem is after a book chapter, then the solution will probably involve material from that chapter, so reread it. If the problem mentions a physical phenomena (e.g. a projectiles) you are most likely supposed to model them, which will result in one or more equations. Etc.

 

[Bliman]

Here you can find a picture of the problem.
https://www.chegg.com/homework-help...-12.p.s-problem-3p-solution-9780618751792-exc
and here is the solution https://www.pricemathteacher.com/uploads/4/0/1/6/40165171/13.4b_projectile_motion.pdf.
I know all the things you are saying and I really try (but it seems like my mind gets numb when finding what I need or how to get to it). But I have big difficulties with these.
I don't only want to point to this problem, it happens in many problems (problem-solving). My mind often turns blank even with all the information stated or when looking at the chapter. It also happens often with proofs. They pull things seemingly out of thin air which I never would come up with. I just don't know how I can get better at it.

 

[HallsofIvy]

"Everyone can plug in numbers and do the number crunching and learn formulas but not everyone can think. "

I sometimes think that, but everyone can learn to think! It takes practice.

A bomber is flying horizontally at an altitude of 3200 feet with a velocity of 400 feet per second when it releases a bomb. A projectile is launched 5 seconds later from a cannon at a site facing the bomber and 5000 feet from the point that was directly beneath the bomber when the bomb was released. The projectile is to intercept the bomb at an altitude of 1600 feet. Determine the required initial speed and angle of inclination of the projectile? (ignore air resistance).

The first thing I would do is set up an xy-coordinate system with the y axis being height. We can take the bomber to be at (0, 3200) with velocity vector (400, 0). The instant the bomb is dropped it has that same position and velocity but also has acceleration (0, -32.2). The bomb's speed, after t seconds, is (400, -32.2t) and its position is (400t, 3200- 16.1t^2).

The cannon is at (5000, 0) if it is aimed at \(\displaystyle \theta\) degrees from the x-axis and has speed v, then its velocity vector is \(\displaystyle (-v cos(\theta), v sin(\theta))\) and its position, after t seconds, is \(\displaystyle 5000- vt cos(\theta), vt sin(\theta)).

In order to intercept the bomb at all, the two positions must be the same at the same time:
\(\displaystyle 400t= 500- vt cos(\theta)\) and \(\displaystyle 3200- 16.1t^2= v sin(\theta)\).

In order that the bullet intercept the bomb at 1600 feet, 3200- 16.1t^2= 1600.

That last equation is easy to solve for t. Putting that value of t into the first two gives two equations to solve for v and \(\displaystyle \theta\).\)

 

[Bliman]

"Everyone can plug in numbers and do the number crunching and learn formulas but not everyone can think. "

I sometimes think that, but everyone can learn to think! It takes practice.

But the question is how? I have always had this problem. It seems like practice doesn't help because every time it is the same.

 

[Bliman]

Does no one know how to get better at it (besides the points stated above)? To me, it looks like practice doesn't really help because every problem is different and needs a different approach.
No books that can help? Does everyone here have a natural ability to do these problems? And if not how did you overcome this hurdle?

 

[jonah2.0]

Beer soaked ramblings and recall follows.

...
The point I am trying to make is that I have great great difficulty with all these problem-solving questions. Sometimes they seem to get things out of thin air. From somewhere I have never thought of. I have so many different problems with it. And I want to get better at it.
Can someone tell me what books are good for that? Or what I must do? Am I not smart enough? It is so frustrating.
It seems like you need another mindset to solve these problems.
So I can use all suggestions I can get to get better at it.
Thank you

There is a strong family resemblance about misdeeds, and if you have all the details of a thousand at your finger ends, it is odd if you can’t unravel the thousand and first.
Sherlock Holmes - A Study in Scarlet

How do problem solvers become good at noticing these tricks?
I've come to believe it comes from getting exposed to (or going through, however which way you see it) a lot of problems & exercises (more than a thousand for some).
That in essence is the trick. I once had a conversation with a bodybuilder who in essence told every young aspiring gym member that there really is no trick or shortcut to having a great physique. One must simply go through the daily grind of sets and reps combined with good diet and discipline until he or she gets the envisioned results.
A.N. Whitehead once said that the purpose of education is not to fill a vessel but to kindle a fire. I believe it should be both. I believe a vessel will eventually reach critical temperature that will start a fire.

 

[Deleted member 4993]

Beer soaked ramblings and recall follows.

There is a strong family resemblance about misdeeds, and if you have all the details of a thousand at your finger ends, it is odd if you can’t unravel the thousand and first.
Sherlock Holmes - A Study in Scarlet

How do problem solvers become good at noticing these tricks?
I've come to believe it comes from getting exposed to (or going through, however which way you see it) a lot of problems & exercises (more than a thousand for some).
That in essence is the trick. I once had a conversation with a bodybuilder who in essence told every young aspiring gym member that there really is no trick or shortcut to having a great physique. One must simply go through the daily grind of sets and reps combined with good diet and discipline until he or she gets the envisioned results.
A.N. Whitehead once said that the purpose of education is not to fill a vessel but to kindle a fire. I believe it should be both. I believe a vessel will eventually reach critical temperature that will start a fire.

My sentiments exactly -very very well said.

My beloved GrandPa used tell me - you have to solve 100s of problems to become a master. If that is not sufficient do 200 - do 500 - do 1000 - there is no short-cut. Unfortunately this journey needs to start from childhood. To be able to shoot like Larry Bird:

wake up 4 AM​

​

Go to Boston Gardens Arena - even before the janitors show up​

​

shoot and practice - free throws, line-drive, sprint - 100s of times everyday - even if you are the perennial MVP of the league!!​

No short cuts!!

 

[Bliman]

I have seen that exercises are very important and can lead to more inside. But for me, problem-solving problems are of a different kind.
When you look in your book you also get examples of the exercises and how to tackle them. That is not the case with problem-solving.
There also seem to be often things that come out of the blue how to solve them.
There seems to be no route or template you can follow. And that is where my brain starts to fade or where I don't seem to have the creativity how to solve them, the same with some proofs.
I then look them up and I sometimes still have difficulty when the answer is not detailed, but I can manage it.
I don't want shortcuts, but I want a hold on. Something from where I can learn and grow in this skill.
More often than not and it doesn't matter and what level (it can also happen in precalculus or even lower) I have to look at the solution how to do these things and I see little progress. It seems like an Achilles heel and I want to do something about it. I just don't know how. Maybe there are books where I can learn this.
I had always had this problem and I am now 43 years old (started to study again the last years to prove to myself that I am not dumb since I got to change course at high school learning biology and science and had to study for cook, that was probably because I later discovered I have Aspergers, I think that was the cause ).

 

[lev888]

I have seen that exercises are very important and can lead to more inside. But for me, problem-solving problems are of a different kind.
When you look in your book you also get examples of the exercises and how to tackle them. That is not the case with problem-solving.
There also seem to be often things that come out of the blue how to solve them.
There seems to be no route or template you can follow. And that is where my brain starts to fade or where I don't seem to have the creativity how to solve them, the same with some proofs.
I then look them up and I sometimes still have difficulty when the answer is not detailed, but I can manage it.
I don't want shortcuts, but I want a hold on. Something from where I can learn and grow in this skill.
More often than not and it doesn't matter and what level (it can also happen in precalculus or even lower) I have to look at the solution how to do these things and I see little progress. It seems like an Achilles heel and I want to do something about it. I just don't know how. Maybe there are books where I can learn this.
I had always had this problem and I am now 43 years old (started to study again the last years to prove to myself that I am not dumb since I got to change course at high school learning biology and science and had to study for cook, that was probably because I later discovered I have Aspergers, I think that was the cause ).

I think posting a few examples (one at a time) with your reasoning (up to the point where you can't continue) would help.

 

[JeffM]

People have written books about how to solve problems. Polya wrote a very neat book called "How to Solve It" about solving math problems. I do not have time to write such a book (and probably do not have the knowledge to do more than begin such a book).

A mistake I see a lot of students make is that they start to do something mathematical before they have seriously thought about the problem. Make sure that you understand the problem before you do anything overtly mechanical. Diagrams, tables, numerical examples are all great ways to start the thinking process. Also abstraction is a useful tool. In your bomber problem, what is its mathematical essence? It is where two curves will intersect. Once you see that abstract aspect, other things follow, like a coordinate system, equations for the curves, mathematical techniques, etc.

Students want to get through problems quickly. This is not a mind set that is going to train you in clarifying problems in a way that you understand. It focuses on the specific rather than the abstract.

 

[Deleted member 4993]

People have written books about how to solve problems. Polya wrote a very neat book called "How to Solve It" about solving math problems. I do not have time to write such a book (and probably do not have the knowledge to do more than begin such a book).

A mistake I see a lot of students make is that they start to do something mathematical before they have seriously thought about the problem. Make sure that you understand the problem before you do anything overtly mechanical. Diagrams, tables, numerical examples are all great ways to start the thinking process. Also abstraction is a useful tool. In your bomber problem, what is its mathematical essence? It is where two curves will intersect. Once you see that abstract aspect, other things follow, like a coordinate system, equations for the curves, mathematical techniques, etc.

Students want to get through problems quickly. This is not a mind set that is going to train you in clarifying problems in a way that you understand. It focuses on the specific rather than the abstract.

I love that book!

That book tells you to figure out the "find" and the "givens" of the problem - explicitly write those down (before even thinking about formulas/equation). When I taught Engineering - I had my students go through this exercise for every HW problem. In a test, I assigned 20% of grade for this excercise.

 

[Bliman]

Thanks for the help. I don't often post here, out of fear of being too stupid to follow along.
Ok for this problem I noticed that it was a projectile motion problem. So I started with the position vector for a projectile.
this got me. 400cos(°)t î + (3200 + 400sin (°)t-16t^2) j .
I then try to set (3200+400sin(°)t-16t^2)=1600. Because that is where the bomb and projectile meet. But that is a no-go because you have two variables. Then I am started doing some substitution but basically, I am stuck then.
When I look at the solution they also have (3200 + 400sin (°)t-16t^2) j but then all of a sudden they write 3200-16t^2=1600.
The 400sin (°)t is gone. The same with 400cos(°)t. It is stated and in the next line, it is 400t. Why I don't know. After that, you have a couple of substitutions that are less of a problem. Although you can't forget that the projectile has launched 5 sec later.
But the moment I see these problems my mind goes blank and when try to do them I try to write the information given. Then I try to take something from the chapter like the position vector for a projectile from the chapter before. If that doesn't work I try to search on and sometimes can set something equal. Like here where I set it equal to 1600. If that doesn't solve it I try to isolate t or something and substitute it in.
It is like I am missing a clear mind to do these things. And this happens with so many problem-solving exercises.
Also thanks for the suggestion of the book (I have seen it many times on Amazon, but never ordered it ).

 

[R.M.]

One tip I share with my students: insert yourself into the world of that problem and watch it happen; think about what you see; picture every word and action described; focus on what IS THERE and also pay attention to what IS NOT THERE; play it over and over again until you fully understand what is happening. Then you can start to piece together a solution. In your bomber example, you mentioned some difficulty understanding the 0 degree angle. Did the plane angle upwards, or launch the bomb upwards before release? No, it flew horizontally the whole time (remember that launch angles are often measured with respect to a horizontal).

Slight tangent, one thing that helped me develop and improve my problem solving ability was by reading fiction books. Being able to picture the world that a writer creates will eventually develop into the words in a math problem "coming alive" in your mind. Many of my students that are excellent problems solvers also read for pleasure; many of my students that struggle mightily with problem solving read next to nothing. Even something as simple as a comic book lets your mind turn static images into actions, which is still good brain exercise. Like previous posters said, it takes time, but it is time well spent.

 

[Bliman]

One tip I share with my students: insert yourself into the world of that problem and watch it happen; think about what you see; picture every word and action described; focus on what IS THERE and also pay attention to what IS NOT THERE; play it over and over again until you fully understand what is happening. Then you can start to piece together a solution. In your bomber example, you mentioned some difficulty understanding the 0 degree angle. Did the plane angle upwards, or launch the bomb upwards before release? No, it flew horizontally the whole time (remember that launch angles are often measured with respect to a horizontal).

Slight tangent, one thing that helped me develop and improve my problem solving ability was by reading fiction books. Being able to picture the world that a writer creates will eventually develop into the words in a math problem "coming alive" in your mind. Many of my students that are excellent problems solvers also read for pleasure; many of my students that struggle mightily with problem solving read next to nothing. Even something as simple as a comic book lets your mind turn static images into actions, which is still good brain exercise. Like previous posters said, it takes time, but it is time well spent.

Thanks for the tip. I have always had that problem you are describing with how to interpret things like the bomb. Maybe it is my Aspergers. In my view, the moment a bomb falls it has already created an angle. I see this in physics as well where I am having difficulties how to interpret when something makes contact or when something is moving. Like here when the bomb is still attached and when it is released.
I never read any fiction books, I read books all the time, but they are science books, philosophy, and others. I like puzzles too.

 

[apple2357]

Its a very interesting question. Can you teach problem solving?

Some people think yes and others think not.

In Polya's book he outlines possible strategies to help you get started- For example, guess and check, look for symmetry, relate it to a question that it looks most similar too.. etc.

However, sometimes you have tried these various things and got nowhere. You post the question on this forum and out of nowhere someone offers a method that would never have occurred to you! So what's going on?

I think there are a number of potential things but the key ones for me are

1. experience - It is more than likely the person who looks like they just pulled a rabbit out of the hat have done lots and lots of different problems before and have ideas because of that. There are people on this forum that are just like this and I never fail to be impressed by them.
2. deeper connected knowledge - having a connected knowledge of mathematics allows you do draw on different areas of mathematics in a problem that doesn't look like it requires this approach.

Another thing that i see in good problem solvers ( I would not consider myself one!) is they don't stop when the problem is over. When they get to the end, they often rethink it, see if they can refine it or find alternative methods. In other words, they think about what they have learnt about problem solving in a reflective way. They may not even do this in a conscious way, its just what interests them.

Just some thoughts to mull over...

 

[apple2357]

As far as your Aspergers goes ( you mentioned it twice), my experience is that this shouldn't be a hindrance. In fact, quite the opposite. If you can get into the problem than a stereotypical aspergers approach might allow you to really get stuck into it in a way someone else might give up. Of course, i am speaking in generalities which might not apply in your case!

 

[Bliman]

Its a very interesting question. Can you teach problem solving?

Some people think yes and others think not.

In Polya's book he outlines possible strategies to help you get started- For example, guess and check, look for symmetry, relate it to a question that it looks most similar too.. etc.

However, sometimes you have tried these various things and got nowhere. You post the question on this forum and out of nowhere someone offers a method that would never have occurred to you! So what's going on?

I think there are a number of potential things but the key ones for me are

1. experience - It is more than likely the person who looks like they just pulled a rabbit out of the hat have done lots and lots of different problems before and have ideas because of that. There are people on this forum that are just like this and I never fail to be impressed by them.
2. deeper connected knowledge - having a connected knowledge of mathematics allows you do draw on different areas of mathematics in a problem that doesn't look like it requires this approach.

Another thing that i see in good problem solvers ( I would not consider myself one!) is they don't stop when the problem is over. When they get to the end, they often rethink it, see if they can refine it or find alternative methods. In other words, they think about what they have learnt about problem solving in a reflective way. They may not even do this in a conscious way, its just what interests them.

Just some thoughts to mull over...

Thank you for the answer. I always doubt if I get what I am learning. And I have a sneaky feeling like I don't get it completely. Like I feel like a fraud sometimes. I then look for other resources or try to read it again (what I am learning not the problem). Often times things stick because I do it so often but the chapters I have done before most often don't stick. And it seems like when you want to do these problem-solving exercises you have to pull from many resources.
Is that normal? Because sometimes they grab from a formula from several chapters back and such.
I also have difficulties with physics and I fear it a bit. I have started not so long ago (I said to myself that I first needed to learn Calculus and I am close to ending the book). Physics relies even more on such problems.
I get that you start to recognize problems if you keep on doing them. But like in this example can someone tell me why the angle is zero?
I know R.M stated that the plane is going horizontal but as soon as the bomb is dropped it makes an angle right? So why start when the bomb hasn't dropped yet?

 

[lev888]

In my view, the moment a bomb falls it has already created an angle.

Before the drop the angle is 0, right? The angle increases from 0 as the bomb starts to fall. So, what was the initial angle? Let's say it's not 0, but x>0. It means that the angle jumped from 0 degrees to x in 0 time. Which is not possible.

 

[Bliman]

Before the drop the angle is 0, right? The angle increases from 0 as the bomb starts to fall. So, what was the initial angle? Let's say it's not 0, but x>0. It means that the angle jumped from 0 degrees to x in 0 time. Which is not possible.

Thanks, that makes sense. These things are sometimes confusing me.