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We should be teaching subjects which will (A) ultimately be useful to as many people as possible and (B) exercise the brain as much as possible. I'm inclined t

We should be teaching subjects which will (A) ultimately be useful to as many people as possible and (B) exercise the brain as much as possible. I'm inclined to think statistics and programming would fulfill these requirements better than calculus. ---------- > Here are a few, understanding and fixing a car, understanding music and playing music, art appreciation, literature and understanding the human condition and on and on. High schoolers already spend a lot of time analyzing literature. I do think they should spend more time with other forms of art as well—why teach only literature criticism, when literature is just one of many art forms? However, this work exercises your brain in a distinctly different way than mathematics, and I do think students should learn both.

Jensson 4y ago
Have you people actually taken a basic statistics course that doesn't require calculus? They are all about memorizing formulas that you plug numbers in to calculate different statistical measures, they don't teach you to understand anything at all. We teach students calculus at that age since teaching them statistics is basically a dead end, we teach statistics to those poor students who will need to calculate statistical measures without understanding them but we should not force every kid to suffer through that boring thing. Calculus is way more interesting, since kids can easily understand it and you can derive all results on your own, statistics is just plug and chug, much worse than calculus ever could be since students aren't ready for it yet.
- Wowfunhappy 4y ago
  > Have you people actually taken a basic statistics course that doesn't require calculus? They are all about memorizing formulas that you plug numbers in to calculate different statistical measures, they don't teach you to understand anything at all. Disclaimer I should have included: I personally took statistics in high school instead of calculus. I can't say what my experience would have been like if I'd known calculus, but loved learning statistics. I don't remember exactly what we did, but I recall it being quite conceptual. Certainly not just a ton of formulas.
- analog31 4y ago
  There is a parallel in college physics. Typically, most colleges have two tracks: "Calc-based" and "non-calc-based" physics. Everybody finds the non-calc-based course to be an utterly bewildering exercise in memorizing formulas. Even the calc based students are baffled by it. The calc based course is widely regarded as easier and more intuitive -- if you also take calc. Stats is the same way. I was a math major, and my college had two stats tracks: "Math stats" and "stats for scientists." The first track was 2 semesters, and we had to prove everything. Of course we used calc. The second track was 1 semester, and was an utterly bewildering exercise in memorizing formulas. I took "math stats," but was then asked to run the discussion section for "stats for scientists." There were things that were utterly intuitive to me, but that I couldn't satisfactorily explain to the students, such as the need for different formulas and methods for discrete and continuous distributions. Freshman economics. The professor bent over backwards to make sense of the formulas related to things like the supply and demand curves, because he couldn't use derivatives. Also, it was 1982, and yes, the professor showed us the Laffer Curve.
  - munchbunny 4y ago
    I had a similar experience, I took stats without calculus just because that was what was available/feasible in college, and eventually I took a more rigorous calculus based probability and stats course in my early 30s. Despite having to relearn my calculus, the difference in conceptual clarity was obvious from even the first lecture because the answer to “why does this work?” has a direct interpretation through calculus that you usually can’t get without calculus. Feels a lot like hiking: as you get more fit you also get access to more fun.
    - analog31 4y ago
      Thinking more about this, my answer to "why learn calculus" is that so many ideas and tools of modern life are explained, and even relate to one another, through calculus. Without calculus, many simple things become unapproachable, or have to be approached by rote learning of formulas and rules. When in grad school, a friend of mine taught a freshman course on modern history, and in the first week he introduced a general outline of calculus. The students were surprised, but he explained that calculus is a thread woven through modern history thanks to the importance of science and technology.
- randomdata 4y ago
  > since kids can easily understand it Maybe with a good teacher. I took calculus courses in high school and college and came out clueless. I could manage the rote work to complete the course, but the big picture was left blank. I revisited it later in life as an adult and gained a fuzzy picture, but it is still not well defined in my mind. According to many comments here some mathematical concepts I am well versed in and use regularly, if not daily, are calculus and that surprises me as they don't seem like anything that was presented in said classes. I dare say that calculus has a marketing problem.
  - dougabug 4y ago
    The first AI generated feature length film I’d want to see would be, “It’s a Wonderful Life, Calculus,” where we get to see what human history would’ve been like if we’d never stumbled across Calculus. So the basic idea at the heart of Calculus is that when you break a problem up into tiny enough pieces, in the right way, the pieces become simpler to analyze and approximations become much more accurate. Like, if I break a complex surface into triangles, if the triangles are small enough, many of the physical properties of interest can be computed using the little triangles. The two main manifestations of this principle are differentiation and integration. With differentiation, the simplification which comes from breaking a function into tiny pieces is that the pieces behave asymptotically like linear functions. For integration, the simplification becomes that the tiny fragments eventually tend to have approximately uniform density, so the mass of the whole body is the sum of the masses of all the tiny bits, each of which is just the volume of the fragment times its density. Differentiation and Integration are in some sense inverses, or opposite sides, of single greater idea. In one dimension, the derivative of the (indefinite) integral of a function equals the original function, and the integral of the derivative of a function is that function plus a constant. Almost inverses of each other, but not quite, and this nuance is where some care is required in developing your understanding of the subject.
    - randomdata 4y ago
      That about matches my fuzzy understanding, but the big picture is the struggle. Like as submitted in a comment here about the venerable for loop being rooted in calculus, it makes sense when it's explicitly pointed out, but I haven't established a framework to draw those connections intuitively. To generalize further, given a solution where the use of calculus is explained then that isolated example is comprehensible, but given a problem there is nothing that sparks a "I know, I can solve this with calculus!" I don't feel the same way about other mathematical disciplines that I have studied, even those of which I have spent far less time studying. Perhaps it's simply harder to understand than other areas of math, but then I think that goes against the idea that is easy for kids to understand.

Jensson 4y ago

Have you people actually taken a basic statistics course that doesn't require calculus? They are all about memorizing formulas that you plug numbers in to calculate different statistical measures, they don't teach you to understand anything at all. We teach students calculus at that age since teaching them statistics is basically a dead end, we teach statistics to those poor students who will need to calculate statistical measures without understanding them but we should not force every kid to suffer through that boring thing. Calculus is way more interesting, since kids can easily understand it and you can derive all results on your own, statistics is just plug and chug, much worse than calculus ever could be since students aren't ready for it yet.

Wowfunhappy 4y ago
> Have you people actually taken a basic statistics course that doesn't require calculus? They are all about memorizing formulas that you plug numbers in to calculate different statistical measures, they don't teach you to understand anything at all. Disclaimer I should have included: I personally took statistics in high school instead of calculus. I can't say what my experience would have been like if I'd known calculus, but loved learning statistics. I don't remember exactly what we did, but I recall it being quite conceptual. Certainly not just a ton of formulas.
analog31 4y ago
There is a parallel in college physics. Typically, most colleges have two tracks: "Calc-based" and "non-calc-based" physics. Everybody finds the non-calc-based course to be an utterly bewildering exercise in memorizing formulas. Even the calc based students are baffled by it. The calc based course is widely regarded as easier and more intuitive -- if you also take calc. Stats is the same way. I was a math major, and my college had two stats tracks: "Math stats" and "stats for scientists." The first track was 2 semesters, and we had to prove everything. Of course we used calc. The second track was 1 semester, and was an utterly bewildering exercise in memorizing formulas. I took "math stats," but was then asked to run the discussion section for "stats for scientists." There were things that were utterly intuitive to me, but that I couldn't satisfactorily explain to the students, such as the need for different formulas and methods for discrete and continuous distributions. Freshman economics. The professor bent over backwards to make sense of the formulas related to things like the supply and demand curves, because he couldn't use derivatives. Also, it was 1982, and yes, the professor showed us the Laffer Curve.
- munchbunny 4y ago
  I had a similar experience, I took stats without calculus just because that was what was available/feasible in college, and eventually I took a more rigorous calculus based probability and stats course in my early 30s. Despite having to relearn my calculus, the difference in conceptual clarity was obvious from even the first lecture because the answer to “why does this work?” has a direct interpretation through calculus that you usually can’t get without calculus. Feels a lot like hiking: as you get more fit you also get access to more fun.
  - analog31 4y ago
    Thinking more about this, my answer to "why learn calculus" is that so many ideas and tools of modern life are explained, and even relate to one another, through calculus. Without calculus, many simple things become unapproachable, or have to be approached by rote learning of formulas and rules. When in grad school, a friend of mine taught a freshman course on modern history, and in the first week he introduced a general outline of calculus. The students were surprised, but he explained that calculus is a thread woven through modern history thanks to the importance of science and technology.
randomdata 4y ago
> since kids can easily understand it Maybe with a good teacher. I took calculus courses in high school and college and came out clueless. I could manage the rote work to complete the course, but the big picture was left blank. I revisited it later in life as an adult and gained a fuzzy picture, but it is still not well defined in my mind. According to many comments here some mathematical concepts I am well versed in and use regularly, if not daily, are calculus and that surprises me as they don't seem like anything that was presented in said classes. I dare say that calculus has a marketing problem.
- dougabug 4y ago
  The first AI generated feature length film I’d want to see would be, “It’s a Wonderful Life, Calculus,” where we get to see what human history would’ve been like if we’d never stumbled across Calculus. So the basic idea at the heart of Calculus is that when you break a problem up into tiny enough pieces, in the right way, the pieces become simpler to analyze and approximations become much more accurate. Like, if I break a complex surface into triangles, if the triangles are small enough, many of the physical properties of interest can be computed using the little triangles. The two main manifestations of this principle are differentiation and integration. With differentiation, the simplification which comes from breaking a function into tiny pieces is that the pieces behave asymptotically like linear functions. For integration, the simplification becomes that the tiny fragments eventually tend to have approximately uniform density, so the mass of the whole body is the sum of the masses of all the tiny bits, each of which is just the volume of the fragment times its density. Differentiation and Integration are in some sense inverses, or opposite sides, of single greater idea. In one dimension, the derivative of the (indefinite) integral of a function equals the original function, and the integral of the derivative of a function is that function plus a constant. Almost inverses of each other, but not quite, and this nuance is where some care is required in developing your understanding of the subject.
  - randomdata 4y ago
    That about matches my fuzzy understanding, but the big picture is the struggle. Like as submitted in a comment here about the venerable for loop being rooted in calculus, it makes sense when it's explicitly pointed out, but I haven't established a framework to draw those connections intuitively. To generalize further, given a solution where the use of calculus is explained then that isolated example is comprehensible, but given a problem there is nothing that sparks a "I know, I can solve this with calculus!" I don't feel the same way about other mathematical disciplines that I have studied, even those of which I have spent far less time studying. Perhaps it's simply harder to understand than other areas of math, but then I think that goes against the idea that is easy for kids to understand.