How do I get better at teaching myself

[pinkfishegg]

I am an undergraduate physics major who just finished her sophomore year of physics and is currently at a research REU. I found that I had a lot of trouble studying by myself during my past year of school : conversations I had in office hours with my professors and exercises we did during class seemed to help but the hours and hours I spend studying never did. I find I tend to learn better through interaction which is an important part of physics but I become overly depending on other people. Right now I am trying to read through a grad level solid state textbook and seem to have the same problem. I retain ideas after going through them with the grad students but seem to retain nothing when reading the book myself. I'm not sure what to do about this but it seems better to fix it now when I don't have the kind of time pressure I do in school.

I'm a task oriented person so I'm good at getting my homework done and research hours in. However I don't always learn everything I need to know from doing my homework. Right now I don't have any homework to do, just experimental tasks to carry out and theory to teach myself with the help of grad students. I think it would help if I bought an undergraduate textbook but I need more advise than that on how to learn. I've looked for lectures online as well but it seems difficult to find good ones that pertain to a specific topic.

 

[chiro]

Hey pinkfishegg and welcome to the forums.

With the fact in mind that research is hard and graduate level science/engineering/math is hard, I think you should put this in perspective.

In terms of learning though, I think finding the right analogies will help you. To do this you need to put things in your own 'native language' because when this is the case, you will be able to break down things and bring them back together with more ease since the language native to you is natural and easier to deal with.

How this is actually done depends on you, how you learn, the subject and other related factors. It might take a combination of practice problems, experience, reading lots of books, talking to people, asking questions to more experienced people, and so on, but if you are aware of your own native language and the analogies to convert one description (i.e. the stuff in your grad level book) to your own native one, then you will be able to juggle these things in your mind with a lot more ease since you are now dealing with something that is natural and not something that you only superficially understand.

The analogies could be physical and visual, they could be in relation to something more symbolic (equations, identities, and so on) or they could relate to some kind of dynamic process that is more or less physical.

Once you know how you best describe things in your own mind, build a bridge to take you from the books description to your own and things will be a lot easier for you.

 

[transphenomen]

I'm not sure if this will work for you, but it does for me. When I read a fact or theorem, I think "How would I explain this to Tony?" I think about how that conversation would go and explain to him in my own words this new concept. Then I imagine him asking a question about it, I would then answer. I have a whole conversation with this imaginary person and I find that I can retain the information much better then if I don't have the conversation.

I actually do this for most things I learn, no matter where I am. Like during a lecture, I would take what I am learning and put it in my own words so I can explain them to Tony when the lecture is over. I don't have time for a full conversation during a lecture, but I can get the first line in most of the time.

 

[pinkfishegg]

Transphenomen, your advise seems helpful, thanks. However, I find after reading a book, I'm often not confident I understand something completely, so I need to talk it with a professor or another student to make sure that I'm on the right tract. I'm not sure if that's bad or just normal. It seems like a lot of people use office hours regularly but some people never use them at all. This seems to especially be the case in math courses where reading the textbook isn't helpful at all and I need to be around other people to get a basic understanding of what's going on. I'm very bad at remembering math steps without conceptually understanding them and some teachers like to only teach the steps without the concepts. Because of this, I feel I am learning but at the rate I need to be learning and I stopped doing well in math courses after Calc II.

Maybe what your saying would be more helpful when reviewing for a test or something. I find that I can have an entire day before a final to study for it, but I never know where to start. If I have the same amount of time spread over a week, I can talk to people, ensure my ideas are right, and have more confidence on the exam. The freshmen level courses I took had a lot more tutoring options. I was hurt in my sophomore level courses because they weren't there and I know they won't be there in my junior level coursework.

Chiro: What do you mean by native language? I'm an American and my native language is English. Do you just mean by own way of explaining things so that I can understand grad level things at an undergraduate level?

 

[homeomorphic]

I retain ideas after going through them with the grad students but seem to retain nothing when reading the book myself.

The problem is that you don't know how to transfer information into your long term memory. When I figured out how to do this, I was shocked that no one had ever told me how and became very disillusioned and cynical about our educational system ever since. What you need is just a little bit of spaced repetition. The way things usually go is that they just give you assignments and the only way they make you review is when you have a test. And then, when the class is over, most people just move on, forgetting most of what they worked so hard to learn, unless something happens to come and reinforce it later on. Some forgetting is probably okay, and a lot of the most important things do end up getting reinforced, but it's not an ideal situation.

Anyway, it's obvious that repetition helps, but it's not obvious what kind of repetition. The less you have to repeat things, the better. But also, repeating things too much is not even optimal for long term retention purposes. You will have to experiment a little bit, but a strategy that is easily remembered is to review something after one minute, then one hour, then one day, then one week, then one month, then one year. Don't take it too literally, but it's a good guideline.

Besides repetition, you need to find ways to make the information memorable. If it appeals to the imagination or emotions, it becomes memorable. Visualization has been proven to increase recall. This is one reason why it is not okay to be as boring as many people are in math and physics today. Lack of respect for aesthetic things, concepts, and visualizations is very harmful to long term retention. Nobody remembers boring things.

I'm very bad at remembering math steps without conceptually understanding them and some teachers like to only teach the steps without the concepts.

So is everyone else. Which is why I am less than happy with the current state of affairs in math and physics education, to put it mildly. Always look around for different sources to see if you can find the best book out there. Also, get to know the best authors. They are your friends.

 

[pinkfishegg]

[

So is everyone else. Which is why I am less than happy with the current state of affairs in math and physics education, to put it mildly. Always look around for different sources to see if you can find the best book out there. Also, get to know the best authors. They are your friends.

Which authors do you suggest? I find the high up physics textbooks tend to go into more detail mathematically but will glaze over physical concepts. I find the lower level books (even down to non-science major physics and astronomy books) will really explain the concepts even if they don't go into much depth. I find I can spend a lot of time using equations without really knowing what they mean. This seems to happen a lot in physics courses. For example, in my last day of freshman e and m my professor asked us to explain maxwell's equations without using any math. No-one in the class could do it.

How can I go about finding physical meaning in these equations? People say that physics is all math but physics describes the natural world and math is just how we describe it. How do I understand physics from these books, not just understand how to do my homework?

It seems like a lot of people are great at learning math without understand the concepts. When I asked other students how they were studying for calc IV they said the methods to do the math were in my notes, and that I just needed to follow them. If you ask them why they do those steps they usually can't answer. This seems to work well for a lot of people but it's not really my learning style.

Besides repetition, you need to find ways to make the information memorable. If it appeals to the imagination or emotions, it becomes memorable. Visualization has been proven to increase recall. This is one reason why it is not okay to be as boring as many people are in math and physics today. Lack of respect for aesthetic things, concepts, and visualizations is very harmful to long term retention. Nobody remembers boring things.

Physicists boring? I've never thought it that way but I know what you mean. Many seem to look down on the arts and social sciences even though these subjects teach you how to communicate. Personally I'm working on a history minor with my physics degree to improve my writing skills and give me a different perspective.

Anyway I do have a lot of trouble learning math and physics when I can't visualize it. 3-D calculus was difficult for my to visualize. I told my calc III and IV professor that and he told me to just follow mechanical steps. I can never remember them if I can't visualize what I'm doing so I ended up being dependent on people who could explain it to me and didn't really have time to learn everything because I couldn't grasp these ideas from my professor or my textbook.

I need to learn how to visualize and grasp these ideas more by myself since there's go to be less and less help the further I go up. I'm just not sure how to do this. Maybe I do just need to read a textbook that I can relate to more.

Your advice on memorization seems very helpful for things I already know how to do, like homework problems that I've solved once but am a little shaky on. I've never quite been sure how to study for the problems on tests and often feel there's no way to study for them.

 

[homeomorphic]

Which authors do you suggest? I find the high up physics textbooks tend to go into more detail mathematically but will glaze over physical concepts. I find the lower level books (even down to non-science major physics and astronomy books) will really explain the concepts even if they don't go into much depth.

I think I am far enough away from you in my field that I don't know if my favorite authors would work for you. There's always Feynman, but he covers limited ground. As I said, you just have to look around for the best explanations at various sources, and by doing that, you not only learn the best explanations, you also learn who your favorite authors are. Another trick you can use to find more good authors, once you have found one, is to use the references from your favorite authors to see which books they might endorse.

I find I can spend a lot of time using equations without really knowing what they mean. This seems to happen a lot in physics courses. For example, in my last day of freshman e and m my professor asked us to explain maxwell's equations without using any math. No-one in the class could do it.

Of course, you will not know the meaning behind every step of every calculation you do, but particularly, E and M had very deep meaning to me when I first studied it. I had a picture in my mind for all the major concepts. Electrodynamics, though, was kind of hard to conceptualize.

How can I go about finding physical meaning in these equations?

The basic idea is that the divergence of a vector field at a point is the flux going through an infinitesimal box at that point. Somewhat similar idea for curl. The details get kind of involved when you put the displacement current in and all that. But I can see it all in my mind's eye. I had to come up with a lot of it myself, with help from various books.

People say that physics is all math but physics describes the natural world and math is just how we describe it.

Some things really can be a challenge to picture. Actually, knowing more math--conceptual math, provides more ways to picture some things in physics.

How do I understand physics from these books, not just understand how to do my homework?

You have to think for yourself. That has to be part of the solution. But if you don't get enough help, it might be hard. So, you have to start by finding the right books that will get you going.

It seems like a lot of people are great at learning math without understand the concepts. When I asked other students how they were studying for calc IV they said the methods to do the math were in my notes, and that I just needed to follow them. If you ask them why they do those steps they usually can't answer. This seems to work well for a lot of people but it's not really my learning style.

Oh, but they AREN'T great at learning math without understanding the concepts. They are just good at cheating the system and cheating themselves. They won't retain much and at the end, all they will have is a grade and not much knowledge.

Physicists boring? I've never thought it that way but I know what you mean. Many seem to look down on the arts and social sciences even though these subjects teach you how to communicate. Personally I'm working on a history minor with my physics degree to improve my writing skills and give me a different perspective.

Well, for me, conceptualizing and visualizing is fun. Calculations are boring, although they have their place. That is what I mean by calling them boring.

Anyway I do have a lot of trouble learning math and physics when I can't visualize it. 3-D calculus was difficult for my to visualize. I told my calc III and IV professor that and he told me to just follow mechanical steps.

Well, there are times when just following mechanical steps might be appropriate, provided you understand why you are doing it, but for him to say that as a general comment without qualification makes me wonder how he ended up as a math professor or even being able to get a PhD.

I can never remember them if I can't visualize what I'm doing so I ended up being dependent on people who could explain it to me and didn't really have time to learn everything because I couldn't grasp these ideas from my professor or my textbook.

You just need to try harder to think about it on your own. But, actually, talking to other people is a really good strategy sometimes. You just don't want to be TOO dependent on it.

I need to learn how to visualize and grasp these ideas more by myself since there's go to be less and less help the further I go up. I'm just not sure how to do this. Maybe I do just need to read a textbook that I can relate to more.

You could try Visual Complex Analysis. That book helped me a lot with visualization. It is something you need to know in physics, too, I guess. There's some really nice physical reasoning in the last few chapters.

Your advice on memorization seems very helpful for things I already know how to do, like homework problems that I've solved once but am a little shaky on. I've never quite been sure how to study for the problems on tests and often feel there's no way to study for them.

Just do a lot of problems and spend some time thinking really hard about the concepts, so that the concepts are all at your fingertips.

Just remember to ask "why" all the time and try to find the answer.

 

[genericusrnme]

What've I've done to help myself is
1. I've used lots of different textbooks to get different points of view
2. I've written 'textbooks', not with the aim of selling or anything but with the aim of improving my own understanding through tedious explanation
3. Made my own problems and worked through them

I too find the way physics is taught most of the time to be pretty bad.
You asked for book recommendations so I'll give you my view favourites;
Landau and Lifgarbagez - Mechanics, Volume 1 of A Course of Theoretical Physics
JJ Sakurai - Modern Quantum Mechanics
Jackson - Classical Electrodynamics

But really I think the best way to apporach physics is from a strong maths background, go back and pick up some maths textbooks, start with some set theory (Bourbaki - Theory of Sets) move onto analysis and linear algebra (Apostol - Mathematical Analysis, Hoffman - Linear Algebra or Roman - Advanced Linear Algebra) then onto manifolds and the likes (there's a million and one books on manifolds in physics or geometry in physics geared towards physicists that are pretty good), once you've got that under your belt you'll find that a lot of the equations you're using make more sense (especially when you go to the quantum world where intuition rapidly drops off).

How can I go about finding physical meaning in these equations? People say that physics is all math but physics describes the natural world and math is just how we describe it. How do I understand physics from these books, not just understand how to do my homework?

You can look for intuitive ideas behind equations but nothing compares with a rigorous understanding of what's going on in the background, asking to understand the physical meaning in equations without understanding the maths behind them is like asking someone to understand a sentence without knowing the words behind them.

Anyway I do have a lot of trouble learning math and physics when I can't visualize it. 3-D calculus was difficult for my to visualize. I told my calc III and IV professor that and he told me to just follow mechanical steps. I can never remember them if I can't visualize what I'm doing so I ended up being dependent on people who could explain it to me and didn't really have time to learn everything because I couldn't grasp these ideas from my professor or my textbook.

If you're not that good at visualising then you may want to try and look into the more rigorous side of things and see WHY these methods you learn in 'physics math' classes are the way they are.

I need to learn how to visualize and grasp these ideas more by myself since there's go to be less and less help the further I go up. I'm just not sure how to do this. Maybe I do just need to read a textbook that I can relate to more.

Again, I'd say learn the rigour behind the equations, as you get to higher and higher levels of physics you're going to find visualisation becomes impossible (eg. infinite dimention rigged hilbert spaces, spinors and most tensor analysis really).That might just be me though, I come from the stand point that physics should be taught starting from the action principle and working down from there and then introducing new things on top of that.

 

[homeomorphic]

If you're not that good at visualising then you may want to try and look into the more rigorous side of things and see WHY these methods you learn in 'physics math' classes are the way they are.

The rigorous side of things has its own pitfalls, similar to the unenlightening thickets of equations. You should always look for the conceptual side of things. It's not always visual, but it often is--or at least semi-visual.

Again, I'd say learn the rigour behind the equations, as you get to higher and higher levels of physics you're going to find visualisation becomes impossible (eg. infinite dimention rigged hilbert spaces, spinors and most tensor analysis really).

I don't find visualizing all those things impossible. The pictures just get more vague and abstract. It can be difficult, though, yes. So, part of it is the non-visual concepts.

 

[genericusrnme]

The rigorous side of things has its own pitfalls, similar to the unenlightening thickets of equations. You should always look for the conceptual side of things. It's not always visual, but it often is--or at least semi-visual.

I don't find visualizing all those things impossible. The pictures just get more vague and abstract. It can be difficult, though, yes. So, part of it is the non-visual concepts.

Yes, I didn't mean to say that you shouldn't try and have some kind of visual picture but I feel that understanding the rigour that goes behind things helps you understand better.
And it gives you the necessary tools to access more advanced material!

 

[chiro]

Chiro: What do you mean by native language? I'm an American and my native language is English. Do you just mean by own way of explaining things so that I can understand grad level things at an undergraduate level?

Not just that.

The language includes any descriptive capacity required for you to not only describe but break things down easily enough so that you can the analysis you need to.

It has nothing to do with written or spoken language: it's your own internal description native to yourself that will allow you to use the descriptions to understand and solve unknown problems later on.

 

[pinkfishegg]

Would you guys recommend buying the Feynman Lectures? There's a book set on amazon.com. I'm debating whether to get Feynmann's book or Landau and Lifshutz's. It might help me to look through mechanics or thermo before I need to take mechanics or thermo next spring.

 

[homeomorphic]

Would you guys recommend buying the Feynman Lectures? There's a book set on amazon.com. I'm debating whether to get Feynmann's book or Landau and Lifshutz's. It might help me to look through mechanics or thermo before I need to take mechanics or thermo next spring.

If money is an issue, I would always check the library first.

Feynman usually provides good intuition. There are Nobel Laureates who began their careers with the Feynman lectures.

One trap you can fall into, though, is concluding that so and so is a good author and therefore everything he says is the best possible explanation. I don't think that's ever the case (Tristan Needham comes close, but has only written one book so far). The good authors just have a higher percentage of good explanations.

 

[marcusl]

Transphenomen, your advise seems helpful, thanks. However, I find after reading a book, I'm often not confident I understand something completely, so I need to talk it with a professor or another student to make sure that I'm on the right tract. I'm not sure if that's bad or just normal. It seems like a lot of people use office hours regularly but some people never use them at all. This seems to especially be the case in math courses where reading the textbook isn't helpful at all and I need to be around other people to get a basic understanding of what's going on. I'm very bad at remembering math steps without conceptually understanding them and some teachers like to only teach the steps without the concepts. Because of this, I feel I am learning but at the rate I need to be learning and I stopped doing well in math courses after Calc II.

You mention reading your books but never mention working problems. Do you work examples as you go and problems at the end of each chapter? Struggling with a problem truly helps master the material, and solving a problem helps you retain the knowledge you've just acquired. If this weren't true for most students, then texts wouldn't include problems and courses wouldn't assign homework.

 

[pinkfishegg]

You mention reading your books but never mention working problems. Do you work examples as you go and problems at the end of each chapter? Struggling with a problem truly helps master the material, and solving a problem helps you retain the knowledge you've just acquired. If this weren't true for most students, then texts wouldn't include problems and courses wouldn't assign homework.

I work though assigned problems in school. I often feel that just solved a math problem, got an answer, but didn't learn any physics. It's not like I'm not learning anything but I'm getting getting a full grasp on the material.

I guess I have a harder time with subjects where I already have some kind of natural intuition from everyday life (like mechanics, circuitry) than the more abstract stuff where I just agree with my teacher and have no reason to second guess myself (like quantuum mechanics, e and m, astrophysics. I also find the latter category more interesting, so I tend to pay attention more.

 

[Nano-Passion]

I'm the exact opposite of you, I learn better by myself than by interaction from others. When I'm by myself, I have all the time I need to get an intuitive feel of the concept. But when I'm with someone else one on and one, I don't always have that time.

Visualization and conceptual understanding is great! But you need to draw the line. Not everything can be visualized, an optimal student should be able to conceptually understand things as well as memorizing the more arbitrary pieces of information in order to follow the logic of the statements and conclusions throughout. At any rate, you should be able to visualize a number of things in three dimensions (though some things are harder than others). There is no excuse for it here, your everyday life is based on the perception of three dimensions. Close your eyes and try really hard to picture things moving in three dimensions. However long it takes, you will find that you can do it.

What is your study habit like? Maybe you gloss over the information too fast. Going through a physics or math book is a lot different than going through a psychology or English book. You need to go very slowly with a pencil and paper in hand. You need to break down the information into little pieces and think about how everything relates. Analyze the relationships of the variables, think how a variable would change if another one is also changed, plug in some numbers, make up your own problems--whatever it takes. Give us more information on how you study so that we can help you out. One thing that really helps me is to write the things down that don't immediately come to my understanding after a couple iterations. There is a number of ways writing things down helps you understand the material, and it provides great notes for review later on.

I work though assigned problems in school. I often feel that just solved a math problem, got an answer, but didn't learn any physics. It's not like I'm not learning anything but I'm getting getting a full grasp on the material.

I guess I have a harder time with subjects where I already have some kind of natural intuition from everyday life (like mechanics, circuitry) than the more abstract stuff where I just agree with my teacher and have no reason to second guess myself (like quantuum mechanics, e and m, astrophysics. I also find the latter category more interesting, so I tend to pay attention more.

How much time do you dedicate to the class?

 

[Nano-Passion]

Again, I'd say learn the rigour behind the equations, as you get to higher and higher levels of physics you're going to find visualisation becomes impossible (eg. infinite dimention rigged hilbert spaces, spinors and most tensor analysis really).That might just be me though, I come from the stand point that physics should be taught starting from the action principle and working down from there and then introducing new things on top of that.

Physics is different than math. If you just simply follow some mathematics without an intuitive feel of the foundation, then you will make a fool of yourself when you present papers with silly mistakes. Successful physics needs intuition, it should be promoted as much as possible for the undergraduate level. The fact that things are impossible to visualize in the graduate level and beyond should not mean that undergraduates should not try to understand undergraduate physics on a conceptual level. Whenever possible, one should always try to understand things on a conceptual level. To me, conceptual understanding is intimate understanding--Feynman & Einstein are good examples of this.

 

[pinkfishegg]

I'm the exact opposite of you, I learn better by myself than by interaction from others. When I'm by myself, I have all the time I need to get an intuitive feel of the concept. But when I'm with someone else one on and one, I don't always have that time.

Visualization and conceptual understanding is great! But you need to draw the line. Not everything can be visualized, an optimal student should be able to conceptually understand things as well as memorizing the more arbitrary pieces of information in order to follow the logic of the statements and conclusions throughout. At any rate, you should be able to visualize a number of things in three dimensions (though some things are harder than others). There is no excuse for it here, your everyday life is based on the perception of three dimensions. Close your eyes and try really hard to picture things moving in three dimensions. However long it takes, you will find that you can do it.

What is your study habit like? Maybe you gloss over the information too fast. Going through a physics or math book is a lot different than going through a psychology or English book. You need to go very slowly with a pencil and paper in hand. You need to break down the information into little pieces and think about how everything relates. Analyze the relationships of the variables, think how a variable would change if another one is also changed, plug in some numbers, make up your own problems--whatever it takes.


Give us more information on how you study so that we can help you out. One thing that really helps me is to write the things down that don't immediately come to my understanding after a couple iterations. There is a number of ways writing things down helps you understand the material, and it provides great notes for review later on.




How much time do you dedicate to the class?

I usually tend to try to dedicate all of my time to my class, which might not be good for an undergraduate because it makes me stressed out, but if I try to take a break from it and relax I feel like I'm wasting time. Last semester I spent 8-12 hours per week being a calculus ta (grading), about 4-6 hours per week in calc IV tutoring (+ time spent alone trying to practice) but when I tried problems on my own I often couldn't get anywhere. Modern, I probably spent about 2-3 hours for every 1 hour outside of class. I did better in that class and learned more than I did in calc IV but I got C's a D,and a high B on the tests. The D may be because I missed a few days of class to attend a physics conference and didn't understand how to deal with Schrodinger's equation until after the test . I also took a history class which I did fine in and a computer science course which i did all my work for in group sessions, but never really grasped any of the concepts. I had no free time last semester, and didn't even take breaks on the weekends, except to go grocery shopping, do laundry, etc.

As far as my study habits go I usually aggressively get all my homework done on it and start it early . My teachers say that my homework solutions are usually well written up and I often get perfect grades on them. However when i need to study for a test I have no idea what to do and and up aimlessly flipping through my notes and skimming through text. Conversations with professors usually help but if I have a test on Monday and I have all weekend to study, I have no idea what to do. In classes where we have no assigned homework or reading, I also have no idea how to study (especially since reading math books is often useless for understanding concepts.) My professors tell me that I'm putting enough time in, I just need to reassess my study habits.

I admit I'm often really distracted when studying by things on the inter net, especially if it's something I'm not that interested into begin with (for example with ac/dc circuits I was really distracted, with modern I was more interested and less distracted.) I also have a hard time working with other students and really getting work done. I find a lot of undergraduate physics majors to be lazy/and or pretentious and its hard to actually work in groups. For example I'll ask another student about calc IV:

me :" I don't understand how to do that problem. Do you get it"
other student:"Yeah he went over it in class, you should know how to do that"
me :"But i don't understand why he used those steps, so I can't remember them"
other student: "Well I never have that problem because I'm good at math"
me :" Uhh, ok can you explain this math then"
other student :"Just follow the steps it's easy"


That's how most student room conversations go. I used to use tutoring more, but I'm going into upper level classes so it doesn't exist anymore. I've tried looking at my math book myself and trying to get the ideas but it often just doesn't click. It did for calc I and II, and math methods in physics but not for Calc III, Calc IV or linear algebra, where I didn't like my teachers or textbooks. I feel I eventually learned Calc III during calc Iv but a lot of linear algebra ideas just didn't sink in. I guess calc I, II, and math methods felt more conceptual which I'm better with. In linear I felt like I was just memorizing definitions and getting proofs thrown at me. In Calc III and IV it felt like a bunch of mechanical steps with no concepts. It's starting to make me dislike math, which is really a bad thing. I try to remind myself that I liked math at one point in time, back when I had teachers I actually liked.

I usually do well on all of my assignments, whether they be a lab report, a paper or a physics homework set. If I'm given enough time I'll work through whatever I need to and hand in good work. However, I usually end up with high C's on a lot of my tests. This isn't really abnormal for physics majors but I'd like to do better. I think it's a problem of not
being able to figure out what's on the test or not knowing what I don't know until its too late. I'm just not sure what to do about it.

 

[Ski900]

I'm not sure if this will work for you, but it does for me. When I read a fact or theorem, I think "How would I explain this to Tony?" I think about how that conversation would go and explain to him in my own words this new concept. Then I imagine him asking a question about it, I would then answer. I have a whole conversation with this imaginary person and I find that I can retain the information much better then if I don't have the conversation.

I actually do this for most things I learn, no matter where I am. Like during a lecture, I would take what I am learning and put it in my own words so I can explain them to Tony when the lecture is over. I don't have time for a full conversation during a lecture, but I can get the first line in most of the time.

I absolutely love this technique! It has helped me out so much. I am currently an EE student and prior to that I spent 11 years in the Navy, so there had been a 12 year gap in any math I had completed or studied. I retaught myself College Algebra and Precalc, and to ensure that I understood every little detail and application of formulas, I would teach what I had learned to my wife (the most patient woman in the world!). This engrained it into my brain. I volunteer as a tutor at my college to help other students out also. It can be time consuming, however, it will help you out leaps and bounds.