Problem solving and so frustrating

[Steven G]

"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.

If I could get a masters degree in math, then yes, everyone can learn to think! Actually the masters degree was easy as I already learned how to think as an undergraduate student. The greatest investment in my like was getting my degrees in math as it taught me how to think clearly. I would never give that up.

 

[Bliman]

If I could get a masters degree in math, then yes, everyone can learn to think! Actually the masters degree was easy as I already learned how to think as an undergraduate student. The greatest investment in my like was getting my degrees in math as it taught me how to think clearly. I would never give that up.

I am a bit confused with this. I try to soak up as much knowledge I can (I am a curious person). Could you tell me how you learned how to think? And if there are books out there that can help me (besides the one suggested in this thread)?
To be honest, I don't believe this, if I could get a master's degree in math, then yes everyone can learn to think. It is nice to think that, but I don't believe it.
I have seen that I don't learn as fast as others and I need to repeat and see it from many different angles to let it sink in.
That is also the reason why I am doing this alone. Because I think if I would take a class then I would be blown away.
But I want to always advance myself. So I am curious how you learned to think?

 

[Steven G]

Before I understand a topic well I too have to look at it from every possible angle. All students of math learn at different rates.

How do you learn how to think? Study math. Do not accept anything that someone tells you as true until you convince yourself of it. I use to tell my students not to believe anything that I tell them, but rather go home and convince yourself that it is true. The main reason that we have so many weak math students is that they accept everything that they are told and just can't remember everything that they are told. If you understand something then you'll never forget it.

I was not much brighter than the average person before starting college. When I started studying math little by little I improved with my thinking ability. I forgot much of what I learned in graduate school, due to the lack of using that material, but I still have the mathematical maturity that I gained from that program. I myself was shocked at how much better I was thinking after my graduate program.

I assure you that if you study abstract math and understand everything you study then you will see an improvement in your thinking. You will need to get to at least Linear Algebra before you start studying abstract math.

 

[Bliman]

Before I understand a topic well I too have to look at it from every possible angle. All students of math learn at different rates.

How do you learn how to think? Study math. Do not accept anything that someone tells you as true until you convince yourself of it. I use to tell my students not to believe anything that I tell them, but rather go home and convince yourself that it is true. The main reason that we have so many weak math students is that they accept everything that they are told and just can't remember everything that they are told. If you understand something then you'll never forget it.

I was not much brighter than the average person before starting college. When I started studying math little by little I improved with my thinking ability. I forgot much of what I learned in graduate school, due to the lack of using that material, but I still have the mathematical maturity that I gained from that program. I myself was shocked at how much better I was thinking after my graduate program.

I assure you that if you study abstract math and understand everything you study then you will see an improvement in your thinking. You will need to get to at least Linear Algebra before you start studying abstract math.

First I want to thank you for your answer.
What I hear is that instead of learning formulas and such and crunch the numbers is to learn the underlying of it all.
I agree with that.
But let's be a little concrete now. We are in the Calculus section. I am near the end of the Larson Calculus book.
How do I convince myself that something is true? How can I really get it? Should I approach this from a different angle?
Because to me, it is a little vague to tell someone to convince yourself that something is true. To do that you have to grasp everything that is stated. But I see no way to do that.
It feels like I am on some sort of Perpetuum mobile. I am close to the end of this book. But I am sure that if I take another Calculus book that I can learn from that as well and I can start again (sure many things I will get but many things I wouldn't). And add to this by then many things I have learned from the previous book have been forgotten.
It seems like chasing a carrot hanging in front of you.
That's why I think you are far too modest of yourself and that you had a gift already that is not given to everyone.
Maybe I need another angle to study Calculus to add to the current one.
You yourself say that when you studied math your thinking ability improved. But you don't say how to achieve this improvement.
Unless you say I need to study abstract math?

 

[nasi112]

First I want to thank you for your answer.
What I hear is that instead of learning formulas and such and crunch the numbers is to learn the underlying of it all.
I agree with that.
But let's be a little concrete now. We are in the Calculus section. I am near the end of the Larson Calculus book.
How do I convince myself that something is true? How can I really get it? Should I approach this from a different angle?
Because to me, it is a little vague to tell someone to convince yourself that something is true. To do that you have to grasp everything that is stated. But I see no way to do that.
It feels like I am on some sort of Perpetuum mobile. I am close to the end of this book. But I am sure that if I take another Calculus book that I can learn from that as well and I can start again (sure many things I will get but many things I wouldn't). And add to this by then many things I have learned from the previous book have been forgotten.
It seems like chasing a carrot hanging in front of you.
That's why I think you are far too modest of yourself and that you had a gift already that is not given to everyone.
Maybe I need another angle to study Calculus to add to the current one.
You yourself say that when you studied math your thinking ability improved. But you don't say how to achieve this improvement.
Unless you say I need to study abstract math?

First, I would like to congratulate you for almost finishing Larson Calculus book. I don't want to disappoint you, but finishing that book is just an introduction to the calculus world. What I want to say is every chapter you have studied in that book has its own book or books. For example, the chapter that talks about the differential equations is not enough to solve many differential equations. You will need to study a book which is only about differential equations and you may spend years in that book if you want to get a solid foundation in solving differential equations. The problem is that when you focus on one topic or book, you will forget the other topics.

Therefore, what is your strategy to master calculus? It seems an impossible and endless journey. Good luck Baliman and forgive me if you feel that I jumped in the middle of nowhere.

 

[lev888]

First I want to thank you for your answer.
What I hear is that instead of learning formulas and such and crunch the numbers is to learn the underlying of it all.
I agree with that.
But let's be a little concrete now. We are in the Calculus section. I am near the end of the Larson Calculus book.
How do I convince myself that something is true? How can I really get it? Should I approach this from a different angle?
Because to me, it is a little vague to tell someone to convince yourself that something is true. To do that you have to grasp everything that is stated. But I see no way to do that.
It feels like I am on some sort of Perpetuum mobile. I am close to the end of this book. But I am sure that if I take another Calculus book that I can learn from that as well and I can start again (sure many things I will get but many things I wouldn't). And add to this by then many things I have learned from the previous book have been forgotten.
It seems like chasing a carrot hanging in front of you.
That's why I think you are far too modest of yourself and that you had a gift already that is not given to everyone.
Maybe I need another angle to study Calculus to add to the current one.
You yourself say that when you studied math your thinking ability improved. But you don't say how to achieve this improvement.
Unless you say I need to study abstract math?

My 2 cents.
Building your math knowledge is similar to building a multi-story house. You can't build the second floor without building the first. If you can't convince yourself that something in Calculus is true it means you have gaps in your knowledge of Algebra or areas of Calculus the book covered previously. If you are studying Calculus in your free time while working, taking care of the family, etc, it's definitely possible to forget some things you studied a while ago. In such cases you need to review the material the current topic references/builds upon.
Regarding thinking ability - not sure what you mean by abstract math, but even arithmetic is already abstract. If you mastered algebra you are way ahead of an average person in abstract thinking. E.g. can a random guy figure out an item's price given the total and the sales tax? You probably can, if you remember how to set up and solve equations.

 

[Steven G]

Do you know what a derivative is? What information does a definite integral give you? Can you now derive the formula for the volume of a cone? How about for the volume of a sphere? A cylinder? Can you explain how to compute the arc length of a curve? How about deriving the integral formulas that have trig inverses as their answers. Does knowing that the derivative of f(x) is positive tell you anything about the concavity of this function?

Please be honest and inform us how many of these questions you can answer.

I can answer all of them. Is it because I am smarter than you? Not necessarily! It is because I made sure that I did not move on until I could answer those questions with affirmative. Stop accepting mathematics. Sure there are somethings that are beyond me and I just accept it but I always try to understand things.

 

[Steven G]

Please be honest and inform us how many of these questions above can you can answer?

 

[jonah2.0]

Beer soaked suggestions follow.

...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 ).

...
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 ).

Not bad for someone who's (probably) a Chef by day and a math hobbyist by night.
Maybe I'll ask you in the future for some tips on how to cook food without salt and still make it appetizing.
As for those books you're asking for, there are a few out there.
For example, at archive.org (The founder of which is presently in litigation against several publishing houses),

How to solve problems; elements of a theory of problems and problem solving : Wickelgren, Wayne A., 1938- : Free Download, Borrow, and Streaming : Internet Archive

How to Solve Problems by Wayne A. Wickelgren ELEMENTS OF A THEORY OF PROBLEMS AND PROBLEM SOLVING from the dedication of the book ….. For as long as I...

archive.org

How to solve problems; elements of a theory of problems and problem solving : Wickelgren, Wayne A., 1938- : Free Download, Borrow, and Streaming : Internet Archive

Bibliography: p. [257]

archive.org

The first link is a community textbook with a PDF download tab.
The 2nd link (same book) can only be viewed/borrowed online by hour or borrowed as a pdf with an expiration date.

 

[jonah2.0]

Beer soaked ramblings follow.
The first and second sentence of the author's Preface should be familiar to you and just about everyone I guess.

 

[JeffM]

Nice to see that Polya was mentioned.

 

[Bliman]

Thanks for all the help.
I shall be completely honest with you guys/girls, I am 43 years old now. When I was in my second year of high school (I think that it is called that way, I live in Belgium). I had to choose another direction. My interests were always science and chemistry and biology and philosophy.
But I had very bad points in math and a few other subjects. Therefore I could only choose a few courses anymore (my sister had learned to cook and therefore I chose that, I wanted to become a baker but I am allergic to flour).
It was a very confusing time for me. And I couldn't really focus on my lectures and it was a very miserable time. I also had many suicidal thoughts.
I had finished successfully as a cook. For me, the theory lessons were fun because it was to easy for me. Now the suicide thoughts stayed. And something had to happen. Sadly I got sacked at my first job very quickly. And that was the nail in the coffin. And I have never recovered (I still depend on government welfare). I have always had a problem with control and pressure. I have been obese and also had/have an eating disorder where I had become anorexic. I have many other mental problems too.
Now to me, intellect is the most important thing for me. I am always someone who changes topics I want to learn and I go deep in them to learn them.
Now when I go to my doctor or psychologist they say that I am very smart, but that doesn't console me because I have nothing to prove that.
The suicide thoughts and depression are still with me. But I vowed to see if Asperger and the many things in high school were the blame of my failure, or that maybe I am just too dumb. In that case, there is no solace for me and I can die with the thought that I indeed am too dumb.

 

[Bliman]

Please be honest and inform us how many of these questions above can you can answer?

I can answer a couple of them. And many I have learned and have forgotten (I don't know if I have to be specific).
What I do feel is that the house is on shaky ground. And I think there lies the crux of the problem. Practice helps me learn things but not always. I want the knowledge I want it to seep deeper and be more grounded. It must feel like I have invented it and can solve it.
So maybe I need other angles so I can tackle Calculus from other sides. If you look at most textbooks you see the same things over and over again. I also see that I struggle when it goes abstract. To make a long story short. It seems like I use the examples in the book of the problems as a guide to solving the problems at the end. And that is not the right way imo. But that is what I want to know.
How can I gain insight? Surely it is not only doing more problems?
Like what should I do when the book is finished. It feels like the knowledge is not sufficiently grounded. I thought let's take another Calculus book and start again. But I am afraid that it is just a waste of time and that the pattern will just repeat itself.

 

[nasi112]

My neighbor has only a primary certificate, but he is 10 times richer than me. He failed to learn what we call sciences and related stuff, but he was very successful in other things. So, learning mathematics and other sciences is not necessary if you are good in other things. You are good at cooking. For me, you are the most successful of all of us (not all, but many) because I failed to cook simple things. Why committing suicide just because you failed at school or having mental issues? School is not everything. Having a family and a sister would let me be happy my whole life even if people think that I am a failure. Keep studying if you are enjoying that and trust in yourself that you are better than millions of people, and thank god a lot that you can walk and breathe. Enjoy your life.

 

[Steven G]

If you like to you can start studying calculus again and after each section you can ask us what you should have learned.
I myself will be hard on you. You say that you learned the product rule. Good! Now can you prove it?! These are the questions which I would ask you.

Now that you have finished calculus you might want to go on to linear algebra or abstract algebra. This will be your first abstract math course. This is where you have to start thinking to understand the material rather than just using some formula.

The only introductory book for abstract algebra that one should use is Topics in Algebra by Herstein. As far as Linear Algebra goes, just stay away from an applied textbook. Kohlman has written a few editions for Linear Algebra which are good. I think the title is Introductory to Linear Algebra. If you really want to play hardball, then use the Linear Algebra book by Hoffman and Kunze. This material will challenge you to think.

If on the other hand you like real analysis then use the book by Rudin, which is commonly called baby Rudin as there is a more advanced version.

Let us know your decision: Calculus, Read Analysis, Linear Algebra or something else.

 

[Bliman]

If you like to you can start studying calculus again and after each section you can ask us what you should have learned.
I myself will be hard on you. You say that you learned the product rule. Good! Now can you prove it?! These are the questions which I would ask you.

Now that you have finished calculus you might want to go on to linear algebra or abstract algebra. This will be your first abstract math course. This is where you have to start thinking to understand the material rather than just using some formula.

The only introductory book for abstract algebra that one should use is Topics in Algebra by Herstein. As far as Linear Algebra goes, just stay away from an applied textbook. Kohlman has written a few editions for Linear Algebra which are good. I think the title is Introductory to Linear Algebra. If you really want to play hardball, then use the Linear Algebra book by Hoffman and Kunze. This material will challenge you to think.

If on the other hand you like real analysis then use the book by Rudin, which is commonly called baby Rudin as there is a more advanced version.

Let us know your decision: Calculus, Read Analysis, Linear Algebra or something else.

I want to thank you.
Your answer really means something to me.
I also expect you to be hard on me. It is the only way to know if I learned it well and deep enough. One thing though. How do you and other people keep the knowledge stored in your brain? Like many things I remember but many things, I forget when things go on. Does it get better if I really deeply know this?
Now I am also at the start of learning physics (I am at the beginning and I am running in the same problems), I follow these steps. https://www.susanjfowler.com/blog/2016/8/13/so-you-want-to-learn-physics ( I am in the beginning of the first physics book). And they say that after the Calculus I should learn this book https://www.amazon.de/-/nl/dp/B08S1TB8Z6/ref=tmm_kin_title_0?_encoding=UTF8&qid=&sr= .
So when I really think about it. It hinges on a few things. You say that for example, you would want me to explain the product rule.
And I know I am a weird guy. But I then would learn the proof of the product rule. And I don't think that is not what you want. I think you want me to know it conceptually too. For that, I need insight and deeper knowledge about it (sort of an aha moment).
Now I would need to post here more often and I can't expect you to be there in the process.
So when I think of it. I would rather do Calculus again, but maybe not an almost repeat but maybe learn it more conceptually. I just don't know what book is good for that. That way I increase my chances of learning it deeper.
So that way I would combine Calculus with the more advanced book above and physics.
Does that sound like a good plan?

 

[Bliman]

My neighbor has only a primary certificate, but he is 10 times richer than me. He failed to learn what we call sciences and related stuff, but he was very successful in other things. So, learning mathematics and other sciences is not necessary if you are good in other things. You are good at cooking. For me, you are the most successful of all of us (not all, but many) because I failed to cook simple things. Why committing suicide just because you failed at school or having mental issues? School is not everything. Having a family and a sister would let me be happy my whole life even if people think that I am a failure. Keep studying if you are enjoying that and trust in yourself that you are better than millions of people, and thank god a lot that you can walk and breathe. Enjoy your life.

Thank you for the boost. Yeah to be honest. To me being good at cooking and such doesn't mean much to me. To me the ultimate is intellect. If you are intelligent then you have the world in your hand. Because you can choose your own destiny.
I don't necessarily want to be rich (enough to not die from starving or cold, and live a decent life).
Imo my life is done, but that is not really a thing to discuss on this forum. I have nothing to be happy about. I am 43 years old. I still live with my parents, I have never ever got a girlfriend, etc... Figuring out who I am and learning every day still gives me some joy in this world. But I found it important that you people know my motivation. I also want to say that math is so beautiful, the same with many things like philosophy and science and psychology and other things. These things I can still cling to.

 

[Steven G]

I want to thank you.
Your answer really means something to me.
I also expect you to be hard on me. It is the only way to know if I learned it well and deep enough. One thing though. How do you and other people keep the knowledge stored in your brain? Like many things I remember but many things, I forget when things go on. Does it get better if I really deeply know this?
Now I am also at the start of learning physics (I am at the beginning and I am running in the same problems), I follow these steps. https://www.susanjfowler.com/blog/2016/8/13/so-you-want-to-learn-physics ( I am in the beginning of the first physics book). And they say that after the Calculus I should learn this book https://www.amazon.de/-/nl/dp/B08S1TB8Z6/ref=tmm_kin_title_0?_encoding=UTF8&qid=&sr= .
So when I really think about it. It hinges on a few things. You say that for example, you would want me to explain the product rule.
And I know I am a weird guy. But I then would learn the proof of the product rule. And I don't think that is not what you want. I think you want me to know it conceptually too. For that, I need insight and deeper knowledge about it (sort of an aha moment).
Now I would need to post here more often and I can't expect you to be there in the process.
So when I think of it. I would rather do Calculus again, but maybe not an almost repeat but maybe learn it more conceptually. I just don't know what book is good for that. That way I increase my chances of learning it deeper.
So that way I would combine Calculus with the more advanced book above and physics.
Does that sound like a good plan?

I did mean that you should know the proof of the product rule! As far as seeing that the product rule is obvious I doubt that one can see it. As I said, you should accept nothing and challenge everything! The proof of the product rule is not that bad, especially if you know what you are looking for (and you do, f'g + fg').

Zill is a very popular differential equation book. If you want the most theoretical calculus book, then you should turn to Thomas and Finn's book. Believe it or not, not Finn and Thomas. The later book is very basic but the former is the most rigorous book out there.

I truly believe that if your goal is to improve thinking, then move away from the mechanical course of calculus and move onto either Real Analysis or Algebra (Linear Algebra or Abstract Algebra). These courses require that you make connections between different definitions and theorems.

I am not sure how much your thinking will improve studying Probability Theory, Combinatorics, Graph Theory, Topology, etc but in my opinion it will be a lower level of thinking then the other topics I included below (maybe Topology should not be in the list I made?)

 

[JeffM]

I am going to disagree (respectfully) with my friend Jomo.

If you want to know how to use calculus, you can memorize some formulas and understand intuitively what types of problems it can deal with. Newton, Leibniz, Euler, and LaGrange are great names in calculus although none of them had a clue about real analysis. None of them could give anything close to a rigorous proof of the product rule. Jomo is asking you to be better at calculus than Newton.

If, on the other hand, you want to be sure that mathematics is logically sound, learn Bourbaki. You are Belgian and so can probably read it easily in French (there may not be a Flemish translation, but there may be a Dutch one).

Moreover, if math is temporarily frustrating you, take a break for a year. Take up a different intellectual interest for a time to let your frustrations dissipate. You could read Henri Pirenne, the only Belgian historian whose works I have read, but who writes like an angel. Maybe sit in a restaurant in Bruges (or Brugge if you prefer), drink a few beers, and read Medieval Cities. There are many things to learn than other derivatives, limits, and integrals.

 

[Bliman]

I did mean that you should know the proof of the product rule! As far as seeing that the product rule is obvious I doubt that one can see it. As I said, you should accept nothing and challenge everything! The proof of the product rule is not that bad, especially if you know what you are looking for (and you do, f'g + fg').

Zill is a very popular differential equation book. If you want the most theoretical calculus book, then you should turn to Thomas and Finn's book. Believe it or not, not Finn and Thomas. The later book is very basic but the former is the most rigorous book out there.

I truly believe that if your goal is to improve thinking, then move away from the mechanical course of calculus and move onto either Real Analysis or Algebra (Linear Algebra or Abstract Algebra). These courses require that you make connections between different definitions and theorems.

I am not sure how much your thinking will improve studying Probability Theory, Combinatorics, Graph Theory, Topology, etc but in my opinion it will be a lower level of thinking then the other topics I included below (maybe Topology should not be in the list I made?)

The first part of your reply I don't get. What does this mean "As far as seeing that the product rule is obvious I doubt that one can see it."? And my goal is to see things more clearly. Rely less on formulas and try to derive them myself. Basically less rote learning and more seeing things myself.
I also don't know if I need the most theoretical calculus book (by the way do you mean this book (https://www.amazon.com/Calculus-Analytic-Geometry-George-Thomas/dp/0201531747 ). To me, it looks like a layer cake. I should build as many layers that they all connect and I can see the whole picture, but I need to get it too. If it all flows over my head then it is useless and I get nothing out of it.
Maybe I need a combination of Polya and something else to build those layers (as I am seeing the same problems pop up in physics).
Seeing that Calculus is basically needed for physics it is essential that I need to know that thoroughly first.
So I would start Calculus again and combine it with the Zill book. Or would you leave the Calculus book and go straight to Linear Algebra? My goal is to totally get (to my heart) Calculus, like the back of my hand, and also to know the physics I pointed out on the website.