And the book: the details are left for the student to work out.
Not to mention even between college-level classes: "The school's requirements say that topic X is optional, so we're going to skip it." (In the next class in the sequence, with a different professor.) "This is explainable because of topic X, which was covered in the previous class."
(Yeah that happened to me...)
Many interesting and useful knowledge were skipped in this way, and a gap occurs between the high school and university. I still remember the first day I took calculus at campus, the professor said, "You probably have already learnt polar coordinates in high school, so I'm going to skip it...Wow, who haven't learnt it before? Well...it's quite easy, you can find some books in the library and learn it by yourself."
i somehow missed out on everything to do with a circle in every class i took and in my senior year in highschool i had to beg my teacher to explain it to me
Exactly this happened to us in the first term of the first year of the maths degree!
Oh, man, right on the nose!
I remember first-year university maths courses mainly just going over high school stuff. I think I did learn conic sections at school but not at university. But I've had other kinds of 'don't worry, you'll learn it later' experiences. I got into the habit of deliberately forgetting all the interesting stuff my big brother told me, so I wouldn't be too bored when I was taught it at school. Then later I realised my brother had been reading up on stuff they didn't teach at school, or had learnt things they taught at his school but not mine. So I forgot all that interesting information and never got it back.
At the first college I went to, there was a series of intro courses, 101-102-103 through (I think) 161-162-163. To get the students more on a single level.
The first one met 5 days per week, others met four, and the "standard" series met 3 days per week, giving the lower classes more time to cover things they needed. The highest course set, in the 160s, used a more theoretical approach. But they all ended with the same material either covered, or (based on the entrance text) already known by the students. That way, the more advanced students didn't have to sit through all the review...
EXACTLY!! (Also, some instructors want to skip over the details of conics... Don't know them. Or, don't have the patience or time to teach them!)
We used to joke about professor's "proofs"--- "it is intuitively obvious even to the dullest of minds that..."
That reminds me of a professor that used to make proofs by puting useful information on the blackboard introduced with "fact:"
Welcome to partial differential equations, class. Today, we'll be studying Euler's method.
I just hope they don't criticize you for asking questions.
Exact reason why I never learned LaPlace transform!!!
Oh, that's why the professors never teach conics. Because they never learned them in first place!
Newton invented calculus at 26. What's taking you so long?
Haha, that's the actual case!
Elementary school Algebra:
>T "The area of a circle is pi r squared. Remember this, lets do some word problems, take out your calculator."
>S "but why is that the formula?"
>T "It's too complicated for now, you'll do that later."
Middle School Geometry:
>S "The year is over and we haven't gone over area & volume formula proofs at all! I still have no proof of the area of a circle!"
>T "Sorry, not in the curriculum, and it is pi r squared."
High School Calculus:
>T "We will be finding volumes of solids using disks today, and as we all know, the area of a circle is pi r squared..."
>S "But how do you know that?"
>T "Why, everyone knows the area of a circle is pi r squared!"
>S "But how do you prove that"
>T "Well, you can just take the integral for polar coordinates"
>S "How do you know that formula, without using the formula for area of circle to find it?"
>T "I think we already did that."
>S "Also the formula doesn't work in Hyperbolic or Elliptic geometries, so it isn't that simple, and a logical geometry proof should suffice, involving limits"
>T "This is not a geometry class, it is Calculus!"
mfw schools do not even show students proofs of area / circumference relationships in circles.
Also, same thing (but worse) with volume / surface area of sphere.
We used Spivak's Calculus, in which he derived the trig formulas starting by defining pi, which makes it trivial to prove that the area of a circle is pi x r^2
This picture is very false to me. Not only it never happened that teachers missed an explanation like this, but on the contrary most of them explained again a lot of what was supposed to be already known.
Happened to me. I took Calc II at one school. We stopped right before conic sections. I transferred to a different school and they started Calc III right after conic sections. I'm in grad school now and I don't know anything about them.