Does it make sense to learn mathematical concepts as you encounter them rather than in a fixed progression? [closed]
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I understand the fundamentals of algrebra, but have very limited knowledge of geometry and trigonometry. I wish to learn calculus at this point.
Is it reasonable to begin learning calculus, and learn these other concepts as I encounter them rather than learning the prerequisites in the standard fixed progression (i.e. Algebra I -> Algebra 2 -> Geometry -> Trigonometry, etc)? How difficult will it be to learn these concepts without the prescribed linear progression?
calculus algebra-precalculus differential-equations soft-question
closed as primarily opinion-based by Matthew Towers, mrtaurho, Cesareo, TheSimpliFire, Don Thousand Dec 25 '18 at 15:38
Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
I understand the fundamentals of algrebra, but have very limited knowledge of geometry and trigonometry. I wish to learn calculus at this point.
Is it reasonable to begin learning calculus, and learn these other concepts as I encounter them rather than learning the prerequisites in the standard fixed progression (i.e. Algebra I -> Algebra 2 -> Geometry -> Trigonometry, etc)? How difficult will it be to learn these concepts without the prescribed linear progression?
calculus algebra-precalculus differential-equations soft-question
closed as primarily opinion-based by Matthew Towers, mrtaurho, Cesareo, TheSimpliFire, Don Thousand Dec 25 '18 at 15:38
Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.
Related question: math.stackexchange.com/questions/42152/…
– littleO
Dec 24 '18 at 7:30
This is basically what I've done since being in pre-algebra in 7th grade. At that time I was getting in to programming video games, and I needed algebra, geometry & trigonometry concepts beyond my current education level, but I only taught myself what I needed to know from a practicality standpoint.
– peaceoutside
Dec 24 '18 at 17:14
1
First you think you understand algebra, then you get to abstract algebra. Then you think you understand calculus, and you get to real analysis. I'm not sure where I'm going with this, except that if you're interested in mathematics try to start proof-heavy work as early as possible in whatever area you find easiest.
– user3067860
Dec 24 '18 at 23:06
add a comment |
I understand the fundamentals of algrebra, but have very limited knowledge of geometry and trigonometry. I wish to learn calculus at this point.
Is it reasonable to begin learning calculus, and learn these other concepts as I encounter them rather than learning the prerequisites in the standard fixed progression (i.e. Algebra I -> Algebra 2 -> Geometry -> Trigonometry, etc)? How difficult will it be to learn these concepts without the prescribed linear progression?
calculus algebra-precalculus differential-equations soft-question
I understand the fundamentals of algrebra, but have very limited knowledge of geometry and trigonometry. I wish to learn calculus at this point.
Is it reasonable to begin learning calculus, and learn these other concepts as I encounter them rather than learning the prerequisites in the standard fixed progression (i.e. Algebra I -> Algebra 2 -> Geometry -> Trigonometry, etc)? How difficult will it be to learn these concepts without the prescribed linear progression?
calculus algebra-precalculus differential-equations soft-question
calculus algebra-precalculus differential-equations soft-question
edited Dec 24 '18 at 6:11
Kemono Chen
2,621437
2,621437
asked Dec 24 '18 at 5:56
Thor Ether
1295
1295
closed as primarily opinion-based by Matthew Towers, mrtaurho, Cesareo, TheSimpliFire, Don Thousand Dec 25 '18 at 15:38
Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as primarily opinion-based by Matthew Towers, mrtaurho, Cesareo, TheSimpliFire, Don Thousand Dec 25 '18 at 15:38
Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.
Related question: math.stackexchange.com/questions/42152/…
– littleO
Dec 24 '18 at 7:30
This is basically what I've done since being in pre-algebra in 7th grade. At that time I was getting in to programming video games, and I needed algebra, geometry & trigonometry concepts beyond my current education level, but I only taught myself what I needed to know from a practicality standpoint.
– peaceoutside
Dec 24 '18 at 17:14
1
First you think you understand algebra, then you get to abstract algebra. Then you think you understand calculus, and you get to real analysis. I'm not sure where I'm going with this, except that if you're interested in mathematics try to start proof-heavy work as early as possible in whatever area you find easiest.
– user3067860
Dec 24 '18 at 23:06
add a comment |
Related question: math.stackexchange.com/questions/42152/…
– littleO
Dec 24 '18 at 7:30
This is basically what I've done since being in pre-algebra in 7th grade. At that time I was getting in to programming video games, and I needed algebra, geometry & trigonometry concepts beyond my current education level, but I only taught myself what I needed to know from a practicality standpoint.
– peaceoutside
Dec 24 '18 at 17:14
1
First you think you understand algebra, then you get to abstract algebra. Then you think you understand calculus, and you get to real analysis. I'm not sure where I'm going with this, except that if you're interested in mathematics try to start proof-heavy work as early as possible in whatever area you find easiest.
– user3067860
Dec 24 '18 at 23:06
Related question: math.stackexchange.com/questions/42152/…
– littleO
Dec 24 '18 at 7:30
Related question: math.stackexchange.com/questions/42152/…
– littleO
Dec 24 '18 at 7:30
This is basically what I've done since being in pre-algebra in 7th grade. At that time I was getting in to programming video games, and I needed algebra, geometry & trigonometry concepts beyond my current education level, but I only taught myself what I needed to know from a practicality standpoint.
– peaceoutside
Dec 24 '18 at 17:14
This is basically what I've done since being in pre-algebra in 7th grade. At that time I was getting in to programming video games, and I needed algebra, geometry & trigonometry concepts beyond my current education level, but I only taught myself what I needed to know from a practicality standpoint.
– peaceoutside
Dec 24 '18 at 17:14
1
1
First you think you understand algebra, then you get to abstract algebra. Then you think you understand calculus, and you get to real analysis. I'm not sure where I'm going with this, except that if you're interested in mathematics try to start proof-heavy work as early as possible in whatever area you find easiest.
– user3067860
Dec 24 '18 at 23:06
First you think you understand algebra, then you get to abstract algebra. Then you think you understand calculus, and you get to real analysis. I'm not sure where I'm going with this, except that if you're interested in mathematics try to start proof-heavy work as early as possible in whatever area you find easiest.
– user3067860
Dec 24 '18 at 23:06
add a comment |
5 Answers
5
active
oldest
votes
Here's a description of how Peter Scholze (a Fields medalist) learns math:
At 16, Scholze learned that a decade earlier Andrew Wiles had proved
the famous 17th-century problem known as Fermat’s Last Theorem, which
says that the equation $x^n + y^n = z^n$ has no nonzero whole-number
solutions if $n$ is greater than two. Scholze was eager to study the
proof, but quickly discovered that despite the problem’s simplicity,
its solution uses some of the most cutting-edge mathematics around. “I
understood nothing, but it was really fascinating,” he said.
So Scholze worked backward, figuring out what he needed to learn to
make sense of the proof. “To this day, that’s to a large extent how I
learn,” he said. “I never really learned the basic things like linear
algebra, actually — I only assimilated it through learning some other
stuff.”
So, I'd say that this backtracking approach to learning math is fine if you enjoy it and it comes naturally to you. But be sure to aim for true, deep understanding of the topics you're interested in rather than rote memorization.
"(...) be sure to aim for true, deep understanding of the topics you're interested in rather than rote memorization." That sentence alone deserves more than a +1. Great example, btw.
– Enzo Ferber
Dec 24 '18 at 12:10
4
+1 even though what works for fields medalists and what works for normal people does not necessarily overlap
– Ant
Dec 25 '18 at 10:01
add a comment |
How difficult will it be to learn these concepts without the prescribed linear progression?
It's not really a linear progression, not at all. All of these subjects actually deal with the same thing. Mathematics is all connected.
Algebra (as it's defined by educators), geometry and trigonometry are taught separately because someone down the line thought it is the best way to teach. It's not. Those subjects just offer different methods/instruments to solve the same problems.
However, you don't really need to know everything that's taught in geometry/trigonometry by heart. Mostly it's all just some particular theorems which are hardly ever used in practice.
What you really need is to have a good grasp on the connections between all of these subjects. For example, all of these complicated trigonometric formulas are derived with the help of algebra together with a small set of rules, connecting $sin$ and $cos$.
Any geometric problem is eventually reduced to an algebraic one, and any algebraic problem (for example, solving an equation) can be represented in geometric terms. Not to mention, that $sin$ and $cos$ are usually first defined through their geometric meaning.
An example: the general solution for a cubic equation can be defined in radicals (with lots of square and cubic roots) or it can be defined in trigonometric form, which looks much neater.
Now, in practice, if you want to study a new subject and are not sure about the prerequisites, just start learning (through textbooks or some other way) and see if you encounter some unclear concepts or methods you don't know. Then use the internet or the help of a tutor to see what you need to learn/remember to understand them.
That's what I do when I need to solve an unfamiliar problem now that I am not longer a student: I just try and search for new methods if needed. It's a fine way to learn, but then again, I do have years of formal education behind me, which helps to recognize the connections I have mentioned.
add a comment |
I'd like to add that learning is not linear no matter what. Even if you learn things in a sequence at the end of the sequence there might be some gain (sometimes huge ones) in revisiting older stuff later on with new conceptual tools and a broader understanding of a field.
There's no obvious path to optimally learn, but I'd say there is at least one mistake that is crucial to avoid: don't hold yourself back thinking that if you don't go deep enough or devote enough time to something than it's not worth looking at. Allow yourself to peek and spend some time in multiple direction. You don't need to learn all there is to know about trigonometry or be highly proficient in using prostaferesis formula to benefit from it. Sit down one afternoon and do a bit of studying and practice some simple trig equation and definition. When you get bored or concerned with time pick up your calculus and go ahead with it. At some point you'll find an integral of a cosine and you will benefit a great deal from just having seen a cos(x) function once. Also, remember that stopping learning a specific tool today doesn't stop you from picking it up again later on.
The more you advance, the more you'll be able to discriminate between when it's time to go deep and what you want to just skim over and use.
add a comment |
Calculus mostly deals with functions: that is what their derivatives are amd what their integrals are.
Whereas trigonometry deals with triangles and angles between two lines, and geometry deals with circles lines, ellipses, again triangles. As these are something one has encountered in life as opposed to functions geometry and trigonometry are easy to see why they could be useful what their applications are.
So it is advisable to study trigonometry and geometry. All illustrations in calculus are mainly using functions of trigonometry and geometry (tangents of curves). In this order understanding would be easier.
add a comment |
Having a personal goal in mind very motivating, and backtracking to learn what you need for specific goals as they come up is fantastic way to learn... as long as you have the time for it.
If you're in a hurry to get something done, though, it can be incredibly frustrating to interrupt your progress for long periods of time.
If you're trying to keep up with your peers in calculus, it can be incredibly frustrating if you have stop, go back, and learn some basics while your peers are pushing forward. Then you have to rush to catch up.
Nonetheless, I think goal-oriented learning is generally much more efficient. The benefit of having a broad base of basic knowledge in a wide variety of subjects is that you never have to backtrack too far.
To answer your specific case: yes, you can learn the fundamentals of calculus without knowing too much about geometry or algebra. You will be at a disadvantage, mostly due to lack of practice on algebra and geometry problems, but it will be easier to work though that than to practice algebra and geometry first with no specific goals in mind.
add a comment |
5 Answers
5
active
oldest
votes
5 Answers
5
active
oldest
votes
active
oldest
votes
active
oldest
votes
Here's a description of how Peter Scholze (a Fields medalist) learns math:
At 16, Scholze learned that a decade earlier Andrew Wiles had proved
the famous 17th-century problem known as Fermat’s Last Theorem, which
says that the equation $x^n + y^n = z^n$ has no nonzero whole-number
solutions if $n$ is greater than two. Scholze was eager to study the
proof, but quickly discovered that despite the problem’s simplicity,
its solution uses some of the most cutting-edge mathematics around. “I
understood nothing, but it was really fascinating,” he said.
So Scholze worked backward, figuring out what he needed to learn to
make sense of the proof. “To this day, that’s to a large extent how I
learn,” he said. “I never really learned the basic things like linear
algebra, actually — I only assimilated it through learning some other
stuff.”
So, I'd say that this backtracking approach to learning math is fine if you enjoy it and it comes naturally to you. But be sure to aim for true, deep understanding of the topics you're interested in rather than rote memorization.
"(...) be sure to aim for true, deep understanding of the topics you're interested in rather than rote memorization." That sentence alone deserves more than a +1. Great example, btw.
– Enzo Ferber
Dec 24 '18 at 12:10
4
+1 even though what works for fields medalists and what works for normal people does not necessarily overlap
– Ant
Dec 25 '18 at 10:01
add a comment |
Here's a description of how Peter Scholze (a Fields medalist) learns math:
At 16, Scholze learned that a decade earlier Andrew Wiles had proved
the famous 17th-century problem known as Fermat’s Last Theorem, which
says that the equation $x^n + y^n = z^n$ has no nonzero whole-number
solutions if $n$ is greater than two. Scholze was eager to study the
proof, but quickly discovered that despite the problem’s simplicity,
its solution uses some of the most cutting-edge mathematics around. “I
understood nothing, but it was really fascinating,” he said.
So Scholze worked backward, figuring out what he needed to learn to
make sense of the proof. “To this day, that’s to a large extent how I
learn,” he said. “I never really learned the basic things like linear
algebra, actually — I only assimilated it through learning some other
stuff.”
So, I'd say that this backtracking approach to learning math is fine if you enjoy it and it comes naturally to you. But be sure to aim for true, deep understanding of the topics you're interested in rather than rote memorization.
"(...) be sure to aim for true, deep understanding of the topics you're interested in rather than rote memorization." That sentence alone deserves more than a +1. Great example, btw.
– Enzo Ferber
Dec 24 '18 at 12:10
4
+1 even though what works for fields medalists and what works for normal people does not necessarily overlap
– Ant
Dec 25 '18 at 10:01
add a comment |
Here's a description of how Peter Scholze (a Fields medalist) learns math:
At 16, Scholze learned that a decade earlier Andrew Wiles had proved
the famous 17th-century problem known as Fermat’s Last Theorem, which
says that the equation $x^n + y^n = z^n$ has no nonzero whole-number
solutions if $n$ is greater than two. Scholze was eager to study the
proof, but quickly discovered that despite the problem’s simplicity,
its solution uses some of the most cutting-edge mathematics around. “I
understood nothing, but it was really fascinating,” he said.
So Scholze worked backward, figuring out what he needed to learn to
make sense of the proof. “To this day, that’s to a large extent how I
learn,” he said. “I never really learned the basic things like linear
algebra, actually — I only assimilated it through learning some other
stuff.”
So, I'd say that this backtracking approach to learning math is fine if you enjoy it and it comes naturally to you. But be sure to aim for true, deep understanding of the topics you're interested in rather than rote memorization.
Here's a description of how Peter Scholze (a Fields medalist) learns math:
At 16, Scholze learned that a decade earlier Andrew Wiles had proved
the famous 17th-century problem known as Fermat’s Last Theorem, which
says that the equation $x^n + y^n = z^n$ has no nonzero whole-number
solutions if $n$ is greater than two. Scholze was eager to study the
proof, but quickly discovered that despite the problem’s simplicity,
its solution uses some of the most cutting-edge mathematics around. “I
understood nothing, but it was really fascinating,” he said.
So Scholze worked backward, figuring out what he needed to learn to
make sense of the proof. “To this day, that’s to a large extent how I
learn,” he said. “I never really learned the basic things like linear
algebra, actually — I only assimilated it through learning some other
stuff.”
So, I'd say that this backtracking approach to learning math is fine if you enjoy it and it comes naturally to you. But be sure to aim for true, deep understanding of the topics you're interested in rather than rote memorization.
answered Dec 24 '18 at 7:39
littleO
29.2k644109
29.2k644109
"(...) be sure to aim for true, deep understanding of the topics you're interested in rather than rote memorization." That sentence alone deserves more than a +1. Great example, btw.
– Enzo Ferber
Dec 24 '18 at 12:10
4
+1 even though what works for fields medalists and what works for normal people does not necessarily overlap
– Ant
Dec 25 '18 at 10:01
add a comment |
"(...) be sure to aim for true, deep understanding of the topics you're interested in rather than rote memorization." That sentence alone deserves more than a +1. Great example, btw.
– Enzo Ferber
Dec 24 '18 at 12:10
4
+1 even though what works for fields medalists and what works for normal people does not necessarily overlap
– Ant
Dec 25 '18 at 10:01
"(...) be sure to aim for true, deep understanding of the topics you're interested in rather than rote memorization." That sentence alone deserves more than a +1. Great example, btw.
– Enzo Ferber
Dec 24 '18 at 12:10
"(...) be sure to aim for true, deep understanding of the topics you're interested in rather than rote memorization." That sentence alone deserves more than a +1. Great example, btw.
– Enzo Ferber
Dec 24 '18 at 12:10
4
4
+1 even though what works for fields medalists and what works for normal people does not necessarily overlap
– Ant
Dec 25 '18 at 10:01
+1 even though what works for fields medalists and what works for normal people does not necessarily overlap
– Ant
Dec 25 '18 at 10:01
add a comment |
How difficult will it be to learn these concepts without the prescribed linear progression?
It's not really a linear progression, not at all. All of these subjects actually deal with the same thing. Mathematics is all connected.
Algebra (as it's defined by educators), geometry and trigonometry are taught separately because someone down the line thought it is the best way to teach. It's not. Those subjects just offer different methods/instruments to solve the same problems.
However, you don't really need to know everything that's taught in geometry/trigonometry by heart. Mostly it's all just some particular theorems which are hardly ever used in practice.
What you really need is to have a good grasp on the connections between all of these subjects. For example, all of these complicated trigonometric formulas are derived with the help of algebra together with a small set of rules, connecting $sin$ and $cos$.
Any geometric problem is eventually reduced to an algebraic one, and any algebraic problem (for example, solving an equation) can be represented in geometric terms. Not to mention, that $sin$ and $cos$ are usually first defined through their geometric meaning.
An example: the general solution for a cubic equation can be defined in radicals (with lots of square and cubic roots) or it can be defined in trigonometric form, which looks much neater.
Now, in practice, if you want to study a new subject and are not sure about the prerequisites, just start learning (through textbooks or some other way) and see if you encounter some unclear concepts or methods you don't know. Then use the internet or the help of a tutor to see what you need to learn/remember to understand them.
That's what I do when I need to solve an unfamiliar problem now that I am not longer a student: I just try and search for new methods if needed. It's a fine way to learn, but then again, I do have years of formal education behind me, which helps to recognize the connections I have mentioned.
add a comment |
How difficult will it be to learn these concepts without the prescribed linear progression?
It's not really a linear progression, not at all. All of these subjects actually deal with the same thing. Mathematics is all connected.
Algebra (as it's defined by educators), geometry and trigonometry are taught separately because someone down the line thought it is the best way to teach. It's not. Those subjects just offer different methods/instruments to solve the same problems.
However, you don't really need to know everything that's taught in geometry/trigonometry by heart. Mostly it's all just some particular theorems which are hardly ever used in practice.
What you really need is to have a good grasp on the connections between all of these subjects. For example, all of these complicated trigonometric formulas are derived with the help of algebra together with a small set of rules, connecting $sin$ and $cos$.
Any geometric problem is eventually reduced to an algebraic one, and any algebraic problem (for example, solving an equation) can be represented in geometric terms. Not to mention, that $sin$ and $cos$ are usually first defined through their geometric meaning.
An example: the general solution for a cubic equation can be defined in radicals (with lots of square and cubic roots) or it can be defined in trigonometric form, which looks much neater.
Now, in practice, if you want to study a new subject and are not sure about the prerequisites, just start learning (through textbooks or some other way) and see if you encounter some unclear concepts or methods you don't know. Then use the internet or the help of a tutor to see what you need to learn/remember to understand them.
That's what I do when I need to solve an unfamiliar problem now that I am not longer a student: I just try and search for new methods if needed. It's a fine way to learn, but then again, I do have years of formal education behind me, which helps to recognize the connections I have mentioned.
add a comment |
How difficult will it be to learn these concepts without the prescribed linear progression?
It's not really a linear progression, not at all. All of these subjects actually deal with the same thing. Mathematics is all connected.
Algebra (as it's defined by educators), geometry and trigonometry are taught separately because someone down the line thought it is the best way to teach. It's not. Those subjects just offer different methods/instruments to solve the same problems.
However, you don't really need to know everything that's taught in geometry/trigonometry by heart. Mostly it's all just some particular theorems which are hardly ever used in practice.
What you really need is to have a good grasp on the connections between all of these subjects. For example, all of these complicated trigonometric formulas are derived with the help of algebra together with a small set of rules, connecting $sin$ and $cos$.
Any geometric problem is eventually reduced to an algebraic one, and any algebraic problem (for example, solving an equation) can be represented in geometric terms. Not to mention, that $sin$ and $cos$ are usually first defined through their geometric meaning.
An example: the general solution for a cubic equation can be defined in radicals (with lots of square and cubic roots) or it can be defined in trigonometric form, which looks much neater.
Now, in practice, if you want to study a new subject and are not sure about the prerequisites, just start learning (through textbooks or some other way) and see if you encounter some unclear concepts or methods you don't know. Then use the internet or the help of a tutor to see what you need to learn/remember to understand them.
That's what I do when I need to solve an unfamiliar problem now that I am not longer a student: I just try and search for new methods if needed. It's a fine way to learn, but then again, I do have years of formal education behind me, which helps to recognize the connections I have mentioned.
How difficult will it be to learn these concepts without the prescribed linear progression?
It's not really a linear progression, not at all. All of these subjects actually deal with the same thing. Mathematics is all connected.
Algebra (as it's defined by educators), geometry and trigonometry are taught separately because someone down the line thought it is the best way to teach. It's not. Those subjects just offer different methods/instruments to solve the same problems.
However, you don't really need to know everything that's taught in geometry/trigonometry by heart. Mostly it's all just some particular theorems which are hardly ever used in practice.
What you really need is to have a good grasp on the connections between all of these subjects. For example, all of these complicated trigonometric formulas are derived with the help of algebra together with a small set of rules, connecting $sin$ and $cos$.
Any geometric problem is eventually reduced to an algebraic one, and any algebraic problem (for example, solving an equation) can be represented in geometric terms. Not to mention, that $sin$ and $cos$ are usually first defined through their geometric meaning.
An example: the general solution for a cubic equation can be defined in radicals (with lots of square and cubic roots) or it can be defined in trigonometric form, which looks much neater.
Now, in practice, if you want to study a new subject and are not sure about the prerequisites, just start learning (through textbooks or some other way) and see if you encounter some unclear concepts or methods you don't know. Then use the internet or the help of a tutor to see what you need to learn/remember to understand them.
That's what I do when I need to solve an unfamiliar problem now that I am not longer a student: I just try and search for new methods if needed. It's a fine way to learn, but then again, I do have years of formal education behind me, which helps to recognize the connections I have mentioned.
answered Dec 24 '18 at 7:48
Yuriy S
15.7k433117
15.7k433117
add a comment |
add a comment |
I'd like to add that learning is not linear no matter what. Even if you learn things in a sequence at the end of the sequence there might be some gain (sometimes huge ones) in revisiting older stuff later on with new conceptual tools and a broader understanding of a field.
There's no obvious path to optimally learn, but I'd say there is at least one mistake that is crucial to avoid: don't hold yourself back thinking that if you don't go deep enough or devote enough time to something than it's not worth looking at. Allow yourself to peek and spend some time in multiple direction. You don't need to learn all there is to know about trigonometry or be highly proficient in using prostaferesis formula to benefit from it. Sit down one afternoon and do a bit of studying and practice some simple trig equation and definition. When you get bored or concerned with time pick up your calculus and go ahead with it. At some point you'll find an integral of a cosine and you will benefit a great deal from just having seen a cos(x) function once. Also, remember that stopping learning a specific tool today doesn't stop you from picking it up again later on.
The more you advance, the more you'll be able to discriminate between when it's time to go deep and what you want to just skim over and use.
add a comment |
I'd like to add that learning is not linear no matter what. Even if you learn things in a sequence at the end of the sequence there might be some gain (sometimes huge ones) in revisiting older stuff later on with new conceptual tools and a broader understanding of a field.
There's no obvious path to optimally learn, but I'd say there is at least one mistake that is crucial to avoid: don't hold yourself back thinking that if you don't go deep enough or devote enough time to something than it's not worth looking at. Allow yourself to peek and spend some time in multiple direction. You don't need to learn all there is to know about trigonometry or be highly proficient in using prostaferesis formula to benefit from it. Sit down one afternoon and do a bit of studying and practice some simple trig equation and definition. When you get bored or concerned with time pick up your calculus and go ahead with it. At some point you'll find an integral of a cosine and you will benefit a great deal from just having seen a cos(x) function once. Also, remember that stopping learning a specific tool today doesn't stop you from picking it up again later on.
The more you advance, the more you'll be able to discriminate between when it's time to go deep and what you want to just skim over and use.
add a comment |
I'd like to add that learning is not linear no matter what. Even if you learn things in a sequence at the end of the sequence there might be some gain (sometimes huge ones) in revisiting older stuff later on with new conceptual tools and a broader understanding of a field.
There's no obvious path to optimally learn, but I'd say there is at least one mistake that is crucial to avoid: don't hold yourself back thinking that if you don't go deep enough or devote enough time to something than it's not worth looking at. Allow yourself to peek and spend some time in multiple direction. You don't need to learn all there is to know about trigonometry or be highly proficient in using prostaferesis formula to benefit from it. Sit down one afternoon and do a bit of studying and practice some simple trig equation and definition. When you get bored or concerned with time pick up your calculus and go ahead with it. At some point you'll find an integral of a cosine and you will benefit a great deal from just having seen a cos(x) function once. Also, remember that stopping learning a specific tool today doesn't stop you from picking it up again later on.
The more you advance, the more you'll be able to discriminate between when it's time to go deep and what you want to just skim over and use.
I'd like to add that learning is not linear no matter what. Even if you learn things in a sequence at the end of the sequence there might be some gain (sometimes huge ones) in revisiting older stuff later on with new conceptual tools and a broader understanding of a field.
There's no obvious path to optimally learn, but I'd say there is at least one mistake that is crucial to avoid: don't hold yourself back thinking that if you don't go deep enough or devote enough time to something than it's not worth looking at. Allow yourself to peek and spend some time in multiple direction. You don't need to learn all there is to know about trigonometry or be highly proficient in using prostaferesis formula to benefit from it. Sit down one afternoon and do a bit of studying and practice some simple trig equation and definition. When you get bored or concerned with time pick up your calculus and go ahead with it. At some point you'll find an integral of a cosine and you will benefit a great deal from just having seen a cos(x) function once. Also, remember that stopping learning a specific tool today doesn't stop you from picking it up again later on.
The more you advance, the more you'll be able to discriminate between when it's time to go deep and what you want to just skim over and use.
answered Dec 24 '18 at 14:58
Three Diag
1907
1907
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Calculus mostly deals with functions: that is what their derivatives are amd what their integrals are.
Whereas trigonometry deals with triangles and angles between two lines, and geometry deals with circles lines, ellipses, again triangles. As these are something one has encountered in life as opposed to functions geometry and trigonometry are easy to see why they could be useful what their applications are.
So it is advisable to study trigonometry and geometry. All illustrations in calculus are mainly using functions of trigonometry and geometry (tangents of curves). In this order understanding would be easier.
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Calculus mostly deals with functions: that is what their derivatives are amd what their integrals are.
Whereas trigonometry deals with triangles and angles between two lines, and geometry deals with circles lines, ellipses, again triangles. As these are something one has encountered in life as opposed to functions geometry and trigonometry are easy to see why they could be useful what their applications are.
So it is advisable to study trigonometry and geometry. All illustrations in calculus are mainly using functions of trigonometry and geometry (tangents of curves). In this order understanding would be easier.
add a comment |
Calculus mostly deals with functions: that is what their derivatives are amd what their integrals are.
Whereas trigonometry deals with triangles and angles between two lines, and geometry deals with circles lines, ellipses, again triangles. As these are something one has encountered in life as opposed to functions geometry and trigonometry are easy to see why they could be useful what their applications are.
So it is advisable to study trigonometry and geometry. All illustrations in calculus are mainly using functions of trigonometry and geometry (tangents of curves). In this order understanding would be easier.
Calculus mostly deals with functions: that is what their derivatives are amd what their integrals are.
Whereas trigonometry deals with triangles and angles between two lines, and geometry deals with circles lines, ellipses, again triangles. As these are something one has encountered in life as opposed to functions geometry and trigonometry are easy to see why they could be useful what their applications are.
So it is advisable to study trigonometry and geometry. All illustrations in calculus are mainly using functions of trigonometry and geometry (tangents of curves). In this order understanding would be easier.
answered Dec 24 '18 at 7:27
P Vanchinathan
14.9k12136
14.9k12136
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Having a personal goal in mind very motivating, and backtracking to learn what you need for specific goals as they come up is fantastic way to learn... as long as you have the time for it.
If you're in a hurry to get something done, though, it can be incredibly frustrating to interrupt your progress for long periods of time.
If you're trying to keep up with your peers in calculus, it can be incredibly frustrating if you have stop, go back, and learn some basics while your peers are pushing forward. Then you have to rush to catch up.
Nonetheless, I think goal-oriented learning is generally much more efficient. The benefit of having a broad base of basic knowledge in a wide variety of subjects is that you never have to backtrack too far.
To answer your specific case: yes, you can learn the fundamentals of calculus without knowing too much about geometry or algebra. You will be at a disadvantage, mostly due to lack of practice on algebra and geometry problems, but it will be easier to work though that than to practice algebra and geometry first with no specific goals in mind.
add a comment |
Having a personal goal in mind very motivating, and backtracking to learn what you need for specific goals as they come up is fantastic way to learn... as long as you have the time for it.
If you're in a hurry to get something done, though, it can be incredibly frustrating to interrupt your progress for long periods of time.
If you're trying to keep up with your peers in calculus, it can be incredibly frustrating if you have stop, go back, and learn some basics while your peers are pushing forward. Then you have to rush to catch up.
Nonetheless, I think goal-oriented learning is generally much more efficient. The benefit of having a broad base of basic knowledge in a wide variety of subjects is that you never have to backtrack too far.
To answer your specific case: yes, you can learn the fundamentals of calculus without knowing too much about geometry or algebra. You will be at a disadvantage, mostly due to lack of practice on algebra and geometry problems, but it will be easier to work though that than to practice algebra and geometry first with no specific goals in mind.
add a comment |
Having a personal goal in mind very motivating, and backtracking to learn what you need for specific goals as they come up is fantastic way to learn... as long as you have the time for it.
If you're in a hurry to get something done, though, it can be incredibly frustrating to interrupt your progress for long periods of time.
If you're trying to keep up with your peers in calculus, it can be incredibly frustrating if you have stop, go back, and learn some basics while your peers are pushing forward. Then you have to rush to catch up.
Nonetheless, I think goal-oriented learning is generally much more efficient. The benefit of having a broad base of basic knowledge in a wide variety of subjects is that you never have to backtrack too far.
To answer your specific case: yes, you can learn the fundamentals of calculus without knowing too much about geometry or algebra. You will be at a disadvantage, mostly due to lack of practice on algebra and geometry problems, but it will be easier to work though that than to practice algebra and geometry first with no specific goals in mind.
Having a personal goal in mind very motivating, and backtracking to learn what you need for specific goals as they come up is fantastic way to learn... as long as you have the time for it.
If you're in a hurry to get something done, though, it can be incredibly frustrating to interrupt your progress for long periods of time.
If you're trying to keep up with your peers in calculus, it can be incredibly frustrating if you have stop, go back, and learn some basics while your peers are pushing forward. Then you have to rush to catch up.
Nonetheless, I think goal-oriented learning is generally much more efficient. The benefit of having a broad base of basic knowledge in a wide variety of subjects is that you never have to backtrack too far.
To answer your specific case: yes, you can learn the fundamentals of calculus without knowing too much about geometry or algebra. You will be at a disadvantage, mostly due to lack of practice on algebra and geometry problems, but it will be easier to work though that than to practice algebra and geometry first with no specific goals in mind.
answered Dec 24 '18 at 16:40
Matt Timmermans
23416
23416
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Related question: math.stackexchange.com/questions/42152/…
– littleO
Dec 24 '18 at 7:30
This is basically what I've done since being in pre-algebra in 7th grade. At that time I was getting in to programming video games, and I needed algebra, geometry & trigonometry concepts beyond my current education level, but I only taught myself what I needed to know from a practicality standpoint.
– peaceoutside
Dec 24 '18 at 17:14
1
First you think you understand algebra, then you get to abstract algebra. Then you think you understand calculus, and you get to real analysis. I'm not sure where I'm going with this, except that if you're interested in mathematics try to start proof-heavy work as early as possible in whatever area you find easiest.
– user3067860
Dec 24 '18 at 23:06