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Do the math: too much calculus? (2012)

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Re: Do the math: too much calculus? (2012)

#131

Post by danielmorozoff »

ghaff wrote: post #98
lordnacho wrote: post #91

Earlier quoted context omitted.

I second that. Stats is so much more applicable, and it is also the area where common sense will fail you the most. Simple things like birthday paradoxes, Gambler's fallacy, Monty Hall problem, and so on are all contrary to common sense yet not taught in schools. And you need stats all the time. Every time someone says crime has never been worse, you need to know how to decide if you believe it. When someone says fly…

I suspect calculus is less commonsensical to most people than you think :-) That said, I've argued the parent's point for years. Basic stats/probability/economic biases are really fundamental stuff and don't require a lot of complicated math if you don't want them to. There are some basic calculus concepts that are useful. Area under a curve. Basic derivatives (for finding maxima/minima, etc.). But the truth is that…

I think the reason calc is preferred, to prob/stats is because the intuition is much more visual than the latter. Furthermore, students up until that point spend the majority of their time studying various functional forms (conic fxns, exps/logs etc.) Functional analysis seems like an obvious next step.

Though stats/prob are perhaps more 'useful' in non-academic life. Imparting the notion of what probability actually is and how Bayes' rule fits into it requires a very gifted teacher with a fundamental grasp of the real mathematics and on top of that the ability to communicate it and link it to previously learned material aka algebra/ pre-calc. You can make it visual, but it would seem to require defining the concepts of sets, sampling, distributions etc.

The issue imo is that teachers tend not to teach the intuitions of math, rather they focus on the mechanics of it. As such, most students turn into machines calculating things as opposed to viewing math as a way to reason.

Re: Do the math: too much calculus? (2012)

#132

Post by lqdc13 »

ivansavz wrote: post #18

I think linear algebra is a much more valuable and general-purpose math tool. I'm not sure how feasible it would be to teach linear algebra topics "properly" to high schoolers, but even a half-assed linear algebra would be more useful than calculus. The connections to geometry, general systems thinking, formal math methods, and countless applications of linear algebra are just the kind of thing students need to get t…

They taught me about matrices and vectors and common matrix/vector operatoins in pre-calculus and even earlier in a shitty public school.

Calculus needs to happen in high school so that people are more prepared to take it again in college. For me, calculus 2 was the hardest class in college. This is coming from someone who went to a math grad school.

Some people need to study way more for calculus than for any other class in high school or college.

Re: Do the math: too much calculus? (2012)

#133

Post by pps43 »

krschultz wrote: post #77

I graduated with a mechanical engineering degree, meaning I took 3 semesters worth of Calc and 3 more semesters worth of differential equations. So I'm not one to hate on math or calculus specifically. But I think high schools should focus more on statistics. Statistics is relevant to more fields than calc. It matters a lot more in business and in all the jobs where you use calc you will also have to use statistics.…

The problem is, it's hard to understand statistics without knowing at least basic calculus. As soon as you move from coin flips to probability density, you need derivatives and integrals.

These days you can solve many practical statistical problems with Monte Carlo, but understanding why those curves look the way they do is still very useful.

Re: Do the math: too much calculus? (2012)

#134

Post by kenjackson »

optionalparens wrote: post #123
krschultz wrote: post #77

I graduated with a mechanical engineering degree, meaning I took 3 semesters worth of Calc and 3 more semesters worth of differential equations. So I'm not one to hate on math or calculus specifically. But I think high schools should focus more on statistics. Statistics is relevant to more fields than calc. It matters a lot more in business and in all the jobs where you use calc you will also have to use statistics.…

One of my degrees is in statistics, and the others are in computer engineering, finance, and computer science. As such I not only took a ton of math, but studied and worked in the contexts you describe where stats is relevant. The problem with statistics education is that it is generally heavily watered down, particularly the classes at the university level for business, psychology, and other majors that need the kno…

Where I went to school there was an undergrad stats sequence for engineers, which was mathematically more intense than the general stats sequence. But it still didn't use much, if any, calculus. And I think in general I'm on board with most here that believe that for most people who aren't going to be engineers, a year of stats is more valuable than a year of calculus.

Re: Do the math: too much calculus? (2012)

#135

Post by danielmorozoff »

pps43 wrote: post #133
krschultz wrote: post #77

I graduated with a mechanical engineering degree, meaning I took 3 semesters worth of Calc and 3 more semesters worth of differential equations. So I'm not one to hate on math or calculus specifically. But I think high schools should focus more on statistics. Statistics is relevant to more fields than calc. It matters a lot more in business and in all the jobs where you use calc you will also have to use statistics.…

The problem is, it's hard to understand statistics without knowing at least basic calculus. As soon as you move from coin flips to probability density, you need derivatives and integrals. These days you can solve many practical statistical problems with Monte Carlo, but understanding why those curves look the way they do is still very useful.

Though i think pdfs are useful, they may be a bit advanced.

Perhaps linking integrals/derivatives into stats could be done from a more discrete approach as opposed to a continuous one first.

For example: Starting with a representation of an integral as sum of areas of rectangles (something most students can do) and then saying what happens if the width of all the rectangles gets smaller and smaller. A la Riemann

This could then tie into the pdf quite reasonably?

Re: Do the math: too much calculus? (2012)

#137

Post by kenjackson »

baldfat wrote: post #6

I think we teach Calculus wrong in a deep fundamental way. We should incorporate more high math concepts earlier into education BUT people are always afraid of anything past 4th grade math! "A minority of students then wend their way through geometry, trigonometry and, finally, calculus, which is considered the pinnacle of high-school-level math. But this progression actually “has nothing to do with how people think,…

"“Calculations kids are forced to do are often so developmentally inappropriate, the experience amounts to torture,” she says. They also miss the essential point—that mathematics is fundamentally about patterns and structures, rather than “little manipulations of numbers,” as she puts it. It’s akin to budding filmmakers learning first about costumes, lighting and other technical aspects, rather than about crafting meaningful stories."

I used to think this until I had kids and started teaching them math. When you know math you can abstract it away and really focus on the concepts. But you don't realize how many simple calculations are done when abstracting these concepts.

Re: Do the math: too much calculus? (2012)

#138

Post by jameshart »

pirocks wrote: post #128
jameshart wrote: post #100

Earlier quoted context omitted.

There's a difference between AP calculus in Alberta and British Columbia? :)

Just in case you didn't know calculus BC is considered the harder AP calculus while AB is considered the easier.

So I went to school in a country that is notorious for forcing kids to specialize early - the UK. In my age cohort, at age sixteen we had to pick just three subjects to study for the next two years. I chose 'pure and applied mathematics' as one of those three subjects. Some people chose 'pure mathematics' and 'applied mathematics' as Two of their subjects. I guess, vaguely, that doing the expanded applied maths would have included more calculus (not sure what that might have entailed - more second order differential equation stuff maybe?) - but there was definitely no point where I was faced with the choice of 'should I do a bit of calculus, or a lot of calculus?'. 16 year olds are not good at knowing how much calculus they need to learn. I was pretty happy just choosing 'do I want to learn some mathematics, no mathematics, or a lot of mathematics?' - and leaving it up to the school to figure out how much calculus to include. Based on my career since, the amount of calculus included was, it turns out, 'enough'.

Re: Do the math: too much calculus? (2012)

#139

Post by amluto »

krschultz wrote: post #77

I graduated with a mechanical engineering degree, meaning I took 3 semesters worth of Calc and 3 more semesters worth of differential equations. So I'm not one to hate on math or calculus specifically. But I think high schools should focus more on statistics. Statistics is relevant to more fields than calc. It matters a lot more in business and in all the jobs where you use calc you will also have to use statistics.…

I would love to see statistics taught more universally. But there's statistics and statistics.

I took a fantastic one-off class as an undergraduate called, IIRC, "Statistics for Physicists". The class taught us Bayes' Law, what tests were, what significance was, and how one might go about deriving and using simple tests. Everyone came out of the class with a very clear understanding of what statistics meant, and no one who took that class would take a statement like "this drug had a significant effect on such-and-such" without an appropriate grain of salt (specifically, how big was the effect). The class didn't include a giant grab-bag of tests.

One of my friends took a regular introductory stats class the following quarter. At one point, he asked me if I could help him do a such-and-such test (Student's t, maybe, but I don't remember which one). I asked him what he was trying to do with the test, and he didn't know.

I think it would be enormously valuable to teach most students what statistics means, what it can do, and what it can't do. I'm not sure how valuable it would be to teach everyone how to evaluate a big pile of magic statistical functions.

Re: Do the math: too much calculus? (2012)

#140

Post by ghaff »

danielmorozoff wrote: post #131
ghaff wrote: post #98

Earlier quoted context omitted.

I suspect calculus is less commonsensical to most people than you think :-) That said, I've argued the parent's point for years. Basic stats/probability/economic biases are really fundamental stuff and don't require a lot of complicated math if you don't want them to. There are some basic calculus concepts that are useful. Area under a curve. Basic derivatives (for finding maxima/minima, etc.). But the truth is that…

I think the reason calc is preferred, to prob/stats is because the intuition is much more visual than the latter. Furthermore, students up until that point spend the majority of their time studying various functional forms (conic fxns, exps/logs etc.) Functional analysis seems like an obvious next step. Though stats/prob are perhaps more 'useful' in non-academic life. Imparting the notion of what probability actually…

>The issue imo is that teachers tend not to teach the intuitions of math, rather they focus on the mechanics of it.

Yes. I agree generally--often even at relatively advanced levels.

The stats course I took in engineering grad school was so heavy on the math that I'm not sure I appreciated what the math was in service of. By contrast when I took a less math-heavy stats course later I much better appreciated the underlying principles.

But yeah, it's very easy to turn any of these math-related topics into: memorize the formula, plug the numbers, come up with a result.

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