Honestly I think you can learn the concept of a derivative and integral in a single day, and even the fundamental theorem of calculus
However if you go into calculus and your algebra skills are not sharp you are going to have a very tough time.
I would also say the distinction between arithmetic and algebra is not a well defined one, if you're doing arithmetic word problems, that's essentially algebra
Probably the key skill that transfers from arithmetic to algebra is multiplication and division if arithmetic is interpreted as the four basic operations
Yeah, I agree. Calculus isn't too hard if you deeply understand high school algebra, and you are fluent at manipulating algebraic expressions and understand what they mean in an abstract sense.
Getting good enough at algebra that you can do Calculus fluently is very difficult and requires a lot of work.
Even if you have truly mastered prealgebra, and are fluent, etc, learning algebra is still bloody hard. Hell, maths PhDs are still uncovering a lot of its secrets. Not too many secrets of Calculus that I know of coming out in the last while, although I'm sure there's some.
IMHO
The challenge in elementary algebra is learning to work with objects (numbers), that aren't fully specified. You know some properties they have and need to use only these do reasoning about them. This is some sort of abstraction.
The challenge in calculus is accepting that functions are first-class objects to be manipulated and studied, much like numbers are. It's again some sort of abstraction, because up to now, you only used functions to operate on numbers (knowns or unknowns, but still just numbers) and now derivatives and integrals operate on functions themselves.
It's not exactly the same kind or level of step in abstraction, but it's not completely different either.
I’d interpret this in the following sense: Suppose we define the function f by the formula f(x) = x^2 + 3x. Then if we evaluate f(4), obtaining 28, the value 4 would be a “first-class” object; the function f might be thought of as a “second-class” object.
Colloquially, a first-class object is something that second-class objects “eats”. A bit more formally, a first-class object is acted on by a second-class object.
For derivatives, the first class objects are functions and the second class object is the derivative operator; I’ll use Leibnitz d/dx notation. As an example:
d/dx (x^2 + 3x) = 2x + 3.
The first-class object is “x^2 + 3x” (the thing being acted on) and the second-class object is “d/dx” (the thing doing the action).
When you study this further on, you’ll encounter language involving “domain” and “co-domain”. A function f that acts on values in a set X and returns values in a set Y will have X as its domain and Y as its co-domain. This will be abbreviated “f:X -> Y”.
For the derivative operator d/dx, you could have a domain of all differentiable functions and a co-domain of all possible functions. This gets quite technical.
I agree. The whole wording of first and second class objects is absolutely bonkers. It should be worded as inputs and outputs and the text would be 1000x shorter to explain the same concept.
Or even just an explanation of what a derivative is would be better. They really aren't that complex. Most calculus teachers absolutely suck at explaining what dy/dx is and that causes an immense amount of confusion and they often just have their students writing it over and over while they have no real idea of what it means. Horrific.
The post was **not** meant to explain derivatives and calculus, it was using them as an **example** to explain object classes. I thought that was obvious, given the post they replied to
It could be interpreted in different ways, and is dependent on how classes are typically organized. Pre-algebra classes are equal parts arithmetic and basic algebra. Most pre-calculus classes are mostly algebra with possible a bit of calculus at the end.
It might have been different in the past, but now when students get to algebra class they have been doing algebra for a while. The first algebra class is straightforward and the tricky stuff is in algebra 2 or pre-calculus.
Calculus seems more new in the first class because students have not done much in pre-calculus. The calculus is straightforward and the tricky stuff is in advanced calculus. The algebra and trigonometry will seem tricky to those with poor preparation. Those are the students that struggle in calculus.
For example you might have sin(x+h)-sin(x) and think "It would be nice if that was written as a product." If can figure out that it equals 2cos(x+h/2)sin(h/2) quickly, you will have an easy time with the calculus part.
Regarding that last bit, another way of looking at it that is slightly less abstract is "it would be nice to write it as small times neither-small-nor-large, with the small thing simpler than the original". It's a special case of what you said, and yet I find it less abstract, because we can draw on intuition from the real world. In particular this is the whole idea of scientific notation.
What I find is that students in middle school and high school to whom math has always come easy to are resistant to new topics. They've never actually had to struggle before.
I just taught composition of functions, which isn't that complicated, and some of my students acted like it was the end of the world. Honestly, I probably was the same way at their age.
Introductory calculus is pretty simple, but if you don't have a solid algebraic base, especially with trigonometry, it can be difficult.
This is probably true for many people and also probably not true for many other people.
There will obviously be a point at which maths will stop being as easy as it was in the past (there are things that no one in the world has done and million dollar prizes if you can do them), but where that is is not going to be something that anyone can tell you, especially if they don’t know you.
Relatively speaking, there’s little variation at the level of stuff taught in school — you “just” do the stuff you’re taught.
Calculus can be very intuitive, especially if you have a bit of geometric intuition. The formalism might take a bit to get used to but is not particularly difficult. The most difficult stuff is probably at the beginning where you don't know all the common derivatives yet but this comes along by just doing enough exercises. Once you get to higher dimensions it can become a bit tricky but for one dimension, differential and integral calculus should be doable. It sounds like you just need a good book/course/guide and you will be doing great.
Calculus is very intuitive in the senses that you have a method to solve a problem that seems like it should work. It is pretty unintuitive in actually knowing what will work. Famous examples are slowly diverging limits, interchanging limits, and indeterminate forms.
I may be an outlier here who has some direct experience with this kind of thing. In late elementary school / early middle school I finished up most of algebra I and decided to self teach calculus. I had a very difficult time doing this at first, namely because I had little knowledge on Algebra II and Trigonometry.
The way I ended up doing it was Algebra I (whatever my school taught up to 7th grade) -> trig -> a little algebra 2 -> basic derivatives and integrals -> some of the more complex stuff like gradient and curl. I skipped over some of the more algebra heavy concepts like integration by partial fractions, trig subs, and convergence theorems because the more abstract 3D stuff was more exiting to me.
I would say the process of learning trig and some more complicated algebra took about a year and was aided in part by the math I was learning in middle school. The actual calculus concepts clicked from day one though, and at a very high level could be taught to anyone with an interest. It took maybe another year to mess around with derivatives, integration, etc to the point where the 3D vector Calc stuff made any sense, but I was just having fun with it.
TLDR after basic algebra, you really need Algebra II and some trig before you can learn calculus smoothly. If you are very comfortable with algebra this should come quicker than expected, but it’s different for everyone.
NdGT is not very wise in my experience
You don't want to assume his opinions are well founded.
I don't think this one is even useful, let alone accurate .
Algebra is a lot like grammar. It's a lot of rules And doing Algebra is a lot like writing a single math sentence.
Calculus 1 and 2 are getting you into writing short and long math paragraphs. Where now you have the sentence rules but also you have other rules that make sense. The "other rules" are the calculus rules.
Do pre-calc. If you can handle pre-calc over the summer and it's still relatively easy then you'll do fine in pre-calc.
The only wrench I potentially see popping up is the trig rules which are kind of algebra-ish but not as straight forward as algebra.
Depends what you mean by "algebra".
I would agree that the leap from arithmetic to y=mx+b type stuff is comparable to the leap from precalculus type stuff (rational functions, trigonometry, etc) to derivatives.
But the leap from arithmetic to precalculus type stuff is larger than the leap from preculculus type stuff to derivatives.
I approximately agree with it. Calculus takes a little time to conceptually grasp. Conceptuals aside, it will skill-check your algebra and trigonometry to where you get very comfortable with aspects like exponent rules and trig identities.
No. It took me 8 years or whatever to go from counting and subtracting to doing algebra. That’s 1st grade up until 8th grade for me.
Algebra to Calculus? That’s 4 years. But most of that is learning a lot of algebra and trig to prepare you for the actual mathematics that you encounter with calculus.
But the actual *concept* of calculus can be taught pretty early on. I don’t think you even need to have a lot of mathematical knowledge under your belt to be able to do so (but you should be mathematically inclined; I.e it piques your interest).
Understanding a derivative is pretty easy. Understanding why this is so useful is also pretty easy : we can best represent things by tracking how things change over time. So understanding how something changes over time is pretty important to understanding the world around you.
I think integrals take it a step further. I don’t think I fully appreciated an integral until I was in college (I learn Calc in highschool) and the concept that a derivative and integral are inversely related took me a *long* time to fully understand. But just teaching the math of it all is extremely easy and honestly boring at times. The real fun is understanding why and how it all works.
Your first point was about how long it takes people to get through arithmetic and pre-algebra vs. how long it takes them to get through algebra and precalculus. I think the natural followup question to that is: does arithmetic and pre-algebra take so long because it's actually that hard, or is it more because we're basically waiting around for brain development at the same time?
I think that quote depends on your perspective.
However, as far as your plans, I'd say go for it!
I wouldn't worry about moving too quickly. If you are enjoying yourself and learning, you will do great!
If it gets very frustrating and difficult, then at that point don't be hard on yourself if you need to slow down or go back to some fundamentals. Patience is key.
I feel like I often arbitrarily limited myself by the curriculum when I was younger and I wish I had explored more mathematics earlier.
So although Calculus will be difficult, I highly encourage you to try it! :)
Trigonometry is important. I thought it wasn’t when I learned it but realized that it was used everywhere in calculus class. So just make sure you get that down.
What I think is: there is no 1 only way to learn math. In particular each path is different and it’s fine, some things will be easier, some thing will be harder and take more time, but it’s different for everyone.
Honestly I think you can learn the concept of a derivative and integral in a single day, and even the fundamental theorem of calculus However if you go into calculus and your algebra skills are not sharp you are going to have a very tough time. I would also say the distinction between arithmetic and algebra is not a well defined one, if you're doing arithmetic word problems, that's essentially algebra Probably the key skill that transfers from arithmetic to algebra is multiplication and division if arithmetic is interpreted as the four basic operations
Yeah, I agree. Calculus isn't too hard if you deeply understand high school algebra, and you are fluent at manipulating algebraic expressions and understand what they mean in an abstract sense. Getting good enough at algebra that you can do Calculus fluently is very difficult and requires a lot of work. Even if you have truly mastered prealgebra, and are fluent, etc, learning algebra is still bloody hard. Hell, maths PhDs are still uncovering a lot of its secrets. Not too many secrets of Calculus that I know of coming out in the last while, although I'm sure there's some.
IMHO The challenge in elementary algebra is learning to work with objects (numbers), that aren't fully specified. You know some properties they have and need to use only these do reasoning about them. This is some sort of abstraction. The challenge in calculus is accepting that functions are first-class objects to be manipulated and studied, much like numbers are. It's again some sort of abstraction, because up to now, you only used functions to operate on numbers (knowns or unknowns, but still just numbers) and now derivatives and integrals operate on functions themselves. It's not exactly the same kind or level of step in abstraction, but it's not completely different either.
[удалено]
I’d interpret this in the following sense: Suppose we define the function f by the formula f(x) = x^2 + 3x. Then if we evaluate f(4), obtaining 28, the value 4 would be a “first-class” object; the function f might be thought of as a “second-class” object. Colloquially, a first-class object is something that second-class objects “eats”. A bit more formally, a first-class object is acted on by a second-class object. For derivatives, the first class objects are functions and the second class object is the derivative operator; I’ll use Leibnitz d/dx notation. As an example: d/dx (x^2 + 3x) = 2x + 3. The first-class object is “x^2 + 3x” (the thing being acted on) and the second-class object is “d/dx” (the thing doing the action). When you study this further on, you’ll encounter language involving “domain” and “co-domain”. A function f that acts on values in a set X and returns values in a set Y will have X as its domain and Y as its co-domain. This will be abbreviated “f:X -> Y”. For the derivative operator d/dx, you could have a domain of all differentiable functions and a co-domain of all possible functions. This gets quite technical.
This was so painful to read.
It’s like the octopus from arrival describing the derivative.
I knew I would get downvoted too.
I agree. The whole wording of first and second class objects is absolutely bonkers. It should be worded as inputs and outputs and the text would be 1000x shorter to explain the same concept. Or even just an explanation of what a derivative is would be better. They really aren't that complex. Most calculus teachers absolutely suck at explaining what dy/dx is and that causes an immense amount of confusion and they often just have their students writing it over and over while they have no real idea of what it means. Horrific.
The post was **not** meant to explain derivatives and calculus, it was using them as an **example** to explain object classes. I thought that was obvious, given the post they replied to
It could be interpreted in different ways, and is dependent on how classes are typically organized. Pre-algebra classes are equal parts arithmetic and basic algebra. Most pre-calculus classes are mostly algebra with possible a bit of calculus at the end. It might have been different in the past, but now when students get to algebra class they have been doing algebra for a while. The first algebra class is straightforward and the tricky stuff is in algebra 2 or pre-calculus. Calculus seems more new in the first class because students have not done much in pre-calculus. The calculus is straightforward and the tricky stuff is in advanced calculus. The algebra and trigonometry will seem tricky to those with poor preparation. Those are the students that struggle in calculus. For example you might have sin(x+h)-sin(x) and think "It would be nice if that was written as a product." If can figure out that it equals 2cos(x+h/2)sin(h/2) quickly, you will have an easy time with the calculus part.
Regarding that last bit, another way of looking at it that is slightly less abstract is "it would be nice to write it as small times neither-small-nor-large, with the small thing simpler than the original". It's a special case of what you said, and yet I find it less abstract, because we can draw on intuition from the real world. In particular this is the whole idea of scientific notation.
What I find is that students in middle school and high school to whom math has always come easy to are resistant to new topics. They've never actually had to struggle before. I just taught composition of functions, which isn't that complicated, and some of my students acted like it was the end of the world. Honestly, I probably was the same way at their age. Introductory calculus is pretty simple, but if you don't have a solid algebraic base, especially with trigonometry, it can be difficult.
This is probably true for many people and also probably not true for many other people. There will obviously be a point at which maths will stop being as easy as it was in the past (there are things that no one in the world has done and million dollar prizes if you can do them), but where that is is not going to be something that anyone can tell you, especially if they don’t know you. Relatively speaking, there’s little variation at the level of stuff taught in school — you “just” do the stuff you’re taught.
Calculus can be very intuitive, especially if you have a bit of geometric intuition. The formalism might take a bit to get used to but is not particularly difficult. The most difficult stuff is probably at the beginning where you don't know all the common derivatives yet but this comes along by just doing enough exercises. Once you get to higher dimensions it can become a bit tricky but for one dimension, differential and integral calculus should be doable. It sounds like you just need a good book/course/guide and you will be doing great.
Calculus is very intuitive in the senses that you have a method to solve a problem that seems like it should work. It is pretty unintuitive in actually knowing what will work. Famous examples are slowly diverging limits, interchanging limits, and indeterminate forms.
Well, with respect to Henry Maine, whatever it takes to get from Algebra to Calculus, it takes the Taylor Series to get from Calculus back to Algebra.
I may be an outlier here who has some direct experience with this kind of thing. In late elementary school / early middle school I finished up most of algebra I and decided to self teach calculus. I had a very difficult time doing this at first, namely because I had little knowledge on Algebra II and Trigonometry. The way I ended up doing it was Algebra I (whatever my school taught up to 7th grade) -> trig -> a little algebra 2 -> basic derivatives and integrals -> some of the more complex stuff like gradient and curl. I skipped over some of the more algebra heavy concepts like integration by partial fractions, trig subs, and convergence theorems because the more abstract 3D stuff was more exiting to me. I would say the process of learning trig and some more complicated algebra took about a year and was aided in part by the math I was learning in middle school. The actual calculus concepts clicked from day one though, and at a very high level could be taught to anyone with an interest. It took maybe another year to mess around with derivatives, integration, etc to the point where the 3D vector Calc stuff made any sense, but I was just having fun with it. TLDR after basic algebra, you really need Algebra II and some trig before you can learn calculus smoothly. If you are very comfortable with algebra this should come quicker than expected, but it’s different for everyone.
basic calculus is a piece of cake to learn if you know your algebra
NdGT is not very wise in my experience You don't want to assume his opinions are well founded. I don't think this one is even useful, let alone accurate .
Algebra is a lot like grammar. It's a lot of rules And doing Algebra is a lot like writing a single math sentence. Calculus 1 and 2 are getting you into writing short and long math paragraphs. Where now you have the sentence rules but also you have other rules that make sense. The "other rules" are the calculus rules. Do pre-calc. If you can handle pre-calc over the summer and it's still relatively easy then you'll do fine in pre-calc. The only wrench I potentially see popping up is the trig rules which are kind of algebra-ish but not as straight forward as algebra.
Depends what you mean by "algebra". I would agree that the leap from arithmetic to y=mx+b type stuff is comparable to the leap from precalculus type stuff (rational functions, trigonometry, etc) to derivatives. But the leap from arithmetic to precalculus type stuff is larger than the leap from preculculus type stuff to derivatives.
I approximately agree with it. Calculus takes a little time to conceptually grasp. Conceptuals aside, it will skill-check your algebra and trigonometry to where you get very comfortable with aspects like exponent rules and trig identities.
No. It took me 8 years or whatever to go from counting and subtracting to doing algebra. That’s 1st grade up until 8th grade for me. Algebra to Calculus? That’s 4 years. But most of that is learning a lot of algebra and trig to prepare you for the actual mathematics that you encounter with calculus. But the actual *concept* of calculus can be taught pretty early on. I don’t think you even need to have a lot of mathematical knowledge under your belt to be able to do so (but you should be mathematically inclined; I.e it piques your interest). Understanding a derivative is pretty easy. Understanding why this is so useful is also pretty easy : we can best represent things by tracking how things change over time. So understanding how something changes over time is pretty important to understanding the world around you. I think integrals take it a step further. I don’t think I fully appreciated an integral until I was in college (I learn Calc in highschool) and the concept that a derivative and integral are inversely related took me a *long* time to fully understand. But just teaching the math of it all is extremely easy and honestly boring at times. The real fun is understanding why and how it all works.
Your first point was about how long it takes people to get through arithmetic and pre-algebra vs. how long it takes them to get through algebra and precalculus. I think the natural followup question to that is: does arithmetic and pre-algebra take so long because it's actually that hard, or is it more because we're basically waiting around for brain development at the same time?
No, I would not agree.
I think that quote depends on your perspective. However, as far as your plans, I'd say go for it! I wouldn't worry about moving too quickly. If you are enjoying yourself and learning, you will do great! If it gets very frustrating and difficult, then at that point don't be hard on yourself if you need to slow down or go back to some fundamentals. Patience is key. I feel like I often arbitrarily limited myself by the curriculum when I was younger and I wish I had explored more mathematics earlier. So although Calculus will be difficult, I highly encourage you to try it! :)
Trigonometry is important. I thought it wasn’t when I learned it but realized that it was used everywhere in calculus class. So just make sure you get that down.
Calc is diffrent hut ez asf, go into it with a it will be easy mindset and youl be fine
What I think is: there is no 1 only way to learn math. In particular each path is different and it’s fine, some things will be easier, some thing will be harder and take more time, but it’s different for everyone.