1. Can you teach statistics without doing calculus first?
It's quite common. We had such a course which was typically taken by social science majors who needed some statistics in their fields. (I never taught it.) The prereq was just a little algebra. A typical text was Mario Triola's "Elementary Statistics" (see here for the table of contents: https://www.amazon.com/Elementary-Statistics-13th-Mario-Trio...). The students used Minitab as part of the course. But we also had prob & stat courses which used calculus.
1. (a) How can you define the exponential function (or the number e) without calculus (limits)?
When I taught college algebra (= review of high school algebra, around two courses below Calc 1) I'd say something like: "There's a number called e which is approximately 2.7118281828459045 ... - it's kind of like pi, an infinite decimal that never repeats. We've seen there are exponential functions like 2^x, and we have an important exponential function called e^x. You're probably wondering why we're using such a weird number in an exponential function. You'll see how it comes up if you go on to take calculus." No one much complained about any of that.
Even a typical calc course skips a lot of the theoretical background - you don't see the background unless you take a course in real analysis. In teaching college math, you're always "starting in the middle", assuming a lot, and in basic courses you're omitting a great deal of the rigor.
(Also, in calculus it's common to define logs first using a definite integral, then discuss inverse functions, then define e^x as the inverse of ln x [so in particular e is just e^1]. You justify the notation by showing ln x and e^x have the properties you "expect" logs and exponentials to have [e.g. ln (a b) = ln a + ln b].)
2. Why not teach statistics/linear algebra/logic in high school?
High school curricula are often strongly determined by state testing, and by the needs of students who are going on to college to have the math that colleges expect. I took courses in math logic and matrix theory in high school, but they were electives; the "big" course for me from the point of view of college admissions was BC calc. I certainly got no college credit for the logic or matrix theory courses.
So e.g. if you tried to do linear algebra or logic in high school it would come at the expense of topics that a state is testing - then your students do poorly (even though they might know some pretty good math) and you and your school get in trouble.
There is more of a movement to teach stat and data analysis in high school and college - things are changing.
3. Why not teach statistics/linear algebra/logic before/in place of calculus in college?
College math departments are heavily service departments. The bulk of the credit hours are in courses taken by majors from other departments. Those departments (science, engineering, business, and so on) tell math departments what their majors need, and math department have to ensure that the math courses suit. The other departments in turn are often required to have majors take certain courses with certain topics by program accrediting agencies. So there can be a lot of inertia there.
1. Can you teach statistics without doing calculus first?
It's quite common. We had such a course which was typically taken by social science majors who needed some statistics in their fields. (I never taught it.) The prereq was just a little algebra. A typical text was Mario Triola's "Elementary Statistics" (see here for the table of contents: https://www.amazon.com/Elementary-Statistics-13th-Mario-Trio...). The students used Minitab as part of the course. But we also had prob & stat courses which used calculus.
1. (a) How can you define the exponential function (or the number e) without calculus (limits)?
When I taught college algebra (= review of high school algebra, around two courses below Calc 1) I'd say something like: "There's a number called e which is approximately 2.7118281828459045 ... - it's kind of like pi, an infinite decimal that never repeats. We've seen there are exponential functions like 2^x, and we have an important exponential function called e^x. You're probably wondering why we're using such a weird number in an exponential function. You'll see how it comes up if you go on to take calculus." No one much complained about any of that.
Even a typical calc course skips a lot of the theoretical background - you don't see the background unless you take a course in real analysis. In teaching college math, you're always "starting in the middle", assuming a lot, and in basic courses you're omitting a great deal of the rigor.
(Also, in calculus it's common to define logs first using a definite integral, then discuss inverse functions, then define e^x as the inverse of ln x [so in particular e is just e^1]. You justify the notation by showing ln x and e^x have the properties you "expect" logs and exponentials to have [e.g. ln (a b) = ln a + ln b].)
2. Why not teach statistics/linear algebra/logic in high school?
High school curricula are often strongly determined by state testing, and by the needs of students who are going on to college to have the math that colleges expect. I took courses in math logic and matrix theory in high school, but they were electives; the "big" course for me from the point of view of college admissions was BC calc. I certainly got no college credit for the logic or matrix theory courses.
So e.g. if you tried to do linear algebra or logic in high school it would come at the expense of topics that a state is testing - then your students do poorly (even though they might know some pretty good math) and you and your school get in trouble.
There is more of a movement to teach stat and data analysis in high school and college - things are changing.
3. Why not teach statistics/linear algebra/logic before/in place of calculus in college?
College math departments are heavily service departments. The bulk of the credit hours are in courses taken by majors from other departments. Those departments (science, engineering, business, and so on) tell math departments what their majors need, and math department have to ensure that the math courses suit. The other departments in turn are often required to have majors take certain courses with certain topics by program accrediting agencies. So there can be a lot of inertia there.