While there is a logical progression to math, because of it's very nature, a "curriculum" is completely unnecessary. On the contrary, it can disrupt the progression, by forcing certain concepts that a person could achieve an understanding of by another path, their own path.
There are certain concepts that must be understood to understand math. There are infinite paths to achieve that understanding.
You also can't teach science without first teaching the scientific method.
Since you are saying this as if it were in opposition to what I said, I suggest you didn't read what I wrote, since this is almost verbatim what I said.
Once you teach the scientific method (which takes an hour to teach, and a lifetime to master) you have done all the "curriculum" required.
If it is a logical progression, no curriculum is required.
There are many possible logical progressions.
If you force a curriculum, you have to get lucky that you have hit on a "best" one (in an average sense), and that one will almost certainly not be the best for everyone.
The word curriculum is just a word used to describe a progression. Do or do not call it that if you choose but if it acts like a duck... If it wasn't logical none of our kids would graduate. My oldest is on high honor role on the advanced schedule. My second is also on high honor role but not in the advanced schedule. Neither one of them could be doing as well as they are with a logically laid out curriculum.
I am very happy that the fixed course of study they are on is working for them. That doesn't mean a curriculum is required for learning, nor does it mean that it is the best path for learning.
I also did very well on the curriculum, at all stages of learning. I believe I would have done a thousand times better on a different path. My best teachers were the ones that had no curriculum (at least it was extremely flexible, catered to the student). The greatest teacher I ever had was so free form in her teaching it was incredible. I will never forget her (5th grade). Her entire class was learning different shit. She was a maestro. She had a greater influence on my thinking than anyone else in my life (except myself) and was one of the primary inspirations for my endeavors in science.
I also have been a teacher (formally and informally, though not on the same level as a full professor/school teacher). I have taught math, physics, chemistry, biology, philosophy, religion, hell, you name it, I've probably taught it (math more than anything). A lot of that was forced by a curriculum. The stuff that was not, but within which I had more latitude, was always more well received (and hopefully remembered) by the student.
While there is a logical progression to math, because of it's very nature, a "curriculum" is completely unnecessary. On the contrary, it can disrupt the progression, by forcing certain concepts that a person could achieve an understanding of by another path, their own path.
There are certain concepts that must be understood to understand math. There are infinite paths to achieve that understanding.
Since you are saying this as if it were in opposition to what I said, I suggest you didn't read what I wrote, since this is almost verbatim what I said.
Once you teach the scientific method (which takes an hour to teach, and a lifetime to master) you have done all the "curriculum" required.
One hour. Done. Now it's time to get creative.
If a curriculum is setup correctly it IS a logical progression.
Disregard the second half of my comment.
If it is a logical progression, no curriculum is required.
There are many possible logical progressions.
If you force a curriculum, you have to get lucky that you have hit on a "best" one (in an average sense), and that one will almost certainly not be the best for everyone.
The word curriculum is just a word used to describe a progression. Do or do not call it that if you choose but if it acts like a duck... If it wasn't logical none of our kids would graduate. My oldest is on high honor role on the advanced schedule. My second is also on high honor role but not in the advanced schedule. Neither one of them could be doing as well as they are with a logically laid out curriculum.
I am very happy that the fixed course of study they are on is working for them. That doesn't mean a curriculum is required for learning, nor does it mean that it is the best path for learning.
I also did very well on the curriculum, at all stages of learning. I believe I would have done a thousand times better on a different path. My best teachers were the ones that had no curriculum (at least it was extremely flexible, catered to the student). The greatest teacher I ever had was so free form in her teaching it was incredible. I will never forget her (5th grade). Her entire class was learning different shit. She was a maestro. She had a greater influence on my thinking than anyone else in my life (except myself) and was one of the primary inspirations for my endeavors in science.
I also have been a teacher (formally and informally, though not on the same level as a full professor/school teacher). I have taught math, physics, chemistry, biology, philosophy, religion, hell, you name it, I've probably taught it (math more than anything). A lot of that was forced by a curriculum. The stuff that was not, but within which I had more latitude, was always more well received (and hopefully remembered) by the student.