The discussion revolves around the challenges and experiences of an MD who is returning to mathematics after many years, specifically focusing on precalculus and the transition to higher mathematics. Participants share their thoughts on the learning process, mistakes made during study, and the implications of balancing a demanding job with academic pursuits.
Participants generally agree that making mistakes is a normal part of learning mathematics and that the OP should not be discouraged. However, there are varying perspectives on the impact of the OP's medical career on their ability to learn higher mathematics, and no consensus is reached on the best approach to integrate math learning with their professional responsibilities.
Some participants note the potential challenges of balancing a demanding job with studying mathematics, while others highlight the importance of learning from errors. There is also mention of the need for practical applications to enhance understanding, but specific methods or resources are not universally agreed upon.
This discussion may be useful for individuals returning to mathematics later in life, particularly those balancing professional responsibilities with academic pursuits, as well as those interested in the learning process and overcoming challenges in mathematics.
FactChecker said:The proofs are not what I had in mind when I mentioned abstraction. Some proofs are just one-time tricks, but others give some deeper insight into how you should think about the subject. It is hard to tell which are which. Some proofs that seem like just a one-time trick keep showing up time and again. They really are fundamental. For a math major, you should definitely get used to the epsilon-delta proofs, proof by contradiction, and proof by induction.
FactChecker said:The proofs are not what I had in mind when I mentioned abstraction. Some proofs are just one-time tricks, but others give some deeper insight into how you should think about the subject. It is hard to tell which are which. Some proofs that seem like just a one-time trick keep showing up time and again. They really are fundamental. For a math major, you should definitely get used to the epsilon-delta proofs, proof by contradiction, and proof by induction.
I agree. In fact, my professors stressed to us, and research has only supported them ever since, that you only learn in math/science from making mistakes. The best way to learn is to try, try, and try again. And everytime you make a mistake, you learn how "not" to do it. You learn what "doesn't" make sense. etc.Mondayman said:Mistakes are part of learning process in math and science. It's extremely difficult to be correct 100% of the time. It's important to make note of what kind of mistakes you are making and how to learn from them.
I remember reading that Robert Oppenheimer was known by his colleagues for getting all his physics correct but getting the wrong constants in his work. Even the big names make mistakes.
You are already dealing with examples of abstraction. When you study "functions" and their properties, that is an abstraction. The function is a mathematical concept that can be used in many real-world applications. You are studying its properties that will hold for any particular application. That concept of abstraction will be carried farther as you go deeper into mathematics.Hsopitalist said:And I'll save my question of "what did you mean, then, by an abstraction?" for this time next year...when I hope to be much further along.
That's what I've been doing wrong all these years. I must try harder to get things wrong a few times before I get them right!grandpa2390 said:... that you only learn in math/science from making mistakes.
If you make a statement like this in a forum full of obsessive/compulsive people (like me), you must expect a response. ;>)grandpa2390 said:you only learn in math/science from making mistakes. The best way to learn is to try, try, and try again. And everytime you make a mistake, you learn how "not" to do it. You learn what "doesn't" make sense. etc.
I think you missed the point I was making. It has nothing to do with Gauss or any other great scientist.FactChecker said:If you make a statement like this in a forum full of obsessive/compulsive people (like me), you must expect a response. ;>)
Gauss is not regarded as a genius because of the mistakes he made. There are two types of mistakes. One type is just an error in mechanical calculations. The other type is a conceptual error. I would not worry much about the calculation errors because that is just trying to compete against calculators or computers. The conceptual errors are the ones to work on. The goal is to get enough intuitive understanding of the concepts that those errors do not occur often.
lol, ok wise guy. if you are getting the problem right on the first time, fine. If you are getting it wrong, don't reach for the answer key, keep working at it until you get it right.PeroK said:That's what I've been doing wrong all these years. I must try harder to get things wrong a few times before I get them right!
grandpa2390 said:After you sat there kicking yourself for 30 minutes or 30 hours trying to figure out where you went wrong solving the problem, the Eureka moment that you earn will stay with you for a long time.
the next time you sit down to start working out a problem, you better believe your bookkeeping will improve. You will be checking and double checking your signs. And if you still miss one, it probably won't take you 30 hours to realize it. checking your signs will be the first thing you do.etotheipi said:What if, after 30 hours, you realize you missed a negative sign on the second line of working...
The lesson I learned is to do any complicated calculations in computer code with many intermediate calculations. That makes it possible to see the step-by-step results and decide if they make sense.grandpa2390 said:the next time you sit down to start working out a problem, you better believe your bookkeeping will improve. You will be checking and double checking your signs. And if you still miss one, it probably won't take you 30 hours to realize it. checking your signs will be the first thing you check.
I don't know. There's merit to that opinion. Spending a ton of time trying to figure out where you went wrong just to discover it was a sign error is no fun. But at the same time, the real world doesn't have answer keys and solution manuals. Learning proper bookkeeping and how to go back and find your mistakes is important.etotheipi said:I remember one of my teachers showing me this study which claims that you are working at the correct level if you have an 85% success rate.
There is often merit to getting stuck and having to puzzle stuff out, but if you get to the point where you're just getting everything wrong, I don't think that's healthy (for motivation or general sanity). And in the case of trivial errors like sign errors, spending lots of time searching for them is a bit of a waste in my opinion. Better just to check it and move on, and avoid all of that frustration.
This is a very good point. Checking every single thing requires a discipline which is a learned skill.grandpa2390 said:Learning proper bookkeeping and how to go back and find your mistakes is important.
or even learning how to recognize when you've made a mistake.
FactChecker said:You are already dealing with examples of abstraction. When you study "functions" and their properties, that is an abstraction. The function is a mathematical concept that can be used in many real-world applications. You are studying its properties that will hold for any particular application. That concept of abstraction will be carried farther as you go deeper into mathematics.
Hsopitalist said:Sweet.
And my copy of Moises part 1 came today! So excited
grandpa2390 said:what's that?
How do you like it so far? The first section can be a bit weird for most newcomers. So maybe you do not like this section, but it gets way better. Have you arrived to the discussion of a parabola, a neat discussion of snells law, and parabolic sector? What I found very cool was the discussion of tangency in calculus.Hsopitalist said:It's a calculus text from 1966 recommended by midget dwarf
MidgetDwarf said:How do you like it so far? The first section can be a bit weird for most newcomers. So maybe you do not like this section, but it gets way better. Have you arrived to the discussion of a parabola, a neat discussion of snells law, and parabolic sector? What I found very cool was the discussion of tangency in calculus.