Do the math: too much calculus? (2012) • Hacker News Board

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

#171

Post by ivansavz » Thu, Oct 20, 2016, 3:48 AM UTC

euroclydon wrote: post #26

I have a 5th and a 6th grader. I'm going to buy them an educational copy of Maple soon, so they can have a beautiful platform to plot 2-D and 3-D graphs, solve equations, do derivatives and integrals, and factor algebraic expressions. I want them to do all this in high school, whether or not they take Calculus, so they can get an intuition for it. Then they can memorize the trig rules in college.

You should also consider SymPy, available as a webapp http://live.sympy.org/ or in jupyter notebook form.

Plotting function graphs is a bit harder (matplotlib) but overall I find SymPy to be much more logical and easy to grok than Maple's weird syntax. Here is a short tutorial on how to do basic math using SymPy: https://minireference.com/static/tutorials/sympy_tutorial.pd...

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

#172

Post by retsibsi » Thu, Oct 20, 2016, 4:24 AM UTC

posterboy wrote: post #154

novembermike wrote: post #141

Earlier quoted context omitted.

The Monty Hall problem is about being given two sets of information. First they give you a 33% shot, then they give you a 50% shot. Take the 50% shot.

No, the second chance is supposed to be better than a 0.5 and I could outline the argument, but I don't trust it.

The key is that Monty Hall is guaranteed to show you a losing door. If that weren't the case -- if he just picked one of the remaining doors at random -- then the fact that he happened to pick a losing door would count as evidence that your door is a winner, leaving you with a 50/50 shot. I think that's one source of intuitive discomfort, and even of lasting confusion if the scenario isn't specified precisely.

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

#173

Post by SiVal » Thu, Oct 20, 2016, 4:28 AM UTC

amluto wrote: post #139

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…

This is an area where I think the usual proper math approach of abstract patterns and theorems and proofs is all wrong for the general public (the ones in a general public school).

They should teach it from the perspective of what problems you are trying to solve, what your options are, and how those options work in practice.

The "in practice" part should be lots of simulations with something like a ChromeBook, so people could really see the ideas working. Create a (simulated) bucket with 70% red balls and 30% blue, then pull samples of various sizes and see how well the sample means match what you already know is the "truth" (70/30). You'll soon have everybody seeing that bigger samples aren't always better, but they tend to be, and they'll see how much better they get as a function of size. Make charts of the improvement. Talk about how good the estimate really NEEDS to be.

Then try some tests where the students DON'T know the "truth" about the population and have to use samples to estimate it. How far off do they think they are. How confident about various intervals, etc.

Then start explaining some of the math that lead to formulas that let you calculate these estimations, sample sizes, confidence intervals, etc.

Have estimation games: you get points for getting an estimate within some delta of the population mean, but you have to pay points to buy your samples. So, how big a sample does your team want to buy?

Then continue with other basic prob/stat ideas such as bayesian cancer tests (does this mean you have cancer or not?) and courtroom dramas (does this mean he's probably guilty or not?), commonly misunderstood situations (ex: Simpson's Paradox), how Gaussian estimators begin to fail when things aren't independent, and all sorts of other introductions to quantitative thinking about real life scenarios.

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

#174

Post by retsibsi » Thu, Oct 20, 2016, 4:46 AM UTC

montyhallstuff wrote: post #165

posterboy wrote: post #159

Earlier quoted context omitted.

Yes, indeed it did for me, as well, but then I'm not trusting my intuition that failed me for the three door version, and I'm desperately trying to prove it wrong, because I'm biased by my first commitment to an answer. I think that commitment bias or what it's called is also an explanation for the misunderstanding in the first place. The argument against the increased chance, as I would use, works both ways and that…

Just in case.... A sample space is the set of all possible outcomes of a random experiment. An event is a subset of a sample space. Let S be the finite sample space with equally likely outcomes in it. Let E be an event. Then the probability of E is the number of outcomes in E over the total number of outcomes in S. Now consider 52 cards in a deck. The sample space of outcomes here are the 52 cards. What is the event…

As I mentioned in another comment, it's important to note that Monty Hall is guaranteed to open a door with a goat behind it (a losing door). If he chose randomly and just happened to reveal a goat, the answer would be different.

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

#175

Post by analog31 » Thu, Oct 20, 2016, 4:52 AM UTC

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.

I had kind of a strange experience. While in college, I worked as a math tutor. Since I had taken one semester of stats, the tutoring center decided I was qualified to run a tutoring session for the non-calculus stats course. I was happy for the money, and had been reasonably successful with other tutoring assignments.

When we got to continuous distributions, we kind of hit a wall. Here's the puzzle. Suppose the data are recorded to two decimal places, and are reasonably bounded, for instance they are between 0 and 100. That's a discrete distribution. All of the data in the examples and problem sets were fixed-point.

I literally couldn't justify to someone who had never taken calculus, why a wholly different kind of distributions with its own collection of formulas, was necessary or useful. The teacher had never explained it either.

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

#176

Post by sidusknight » Thu, Oct 20, 2016, 5:14 AM UTC

lqdc13 wrote: post #132

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…

I'm guessing you did applied math in grad school? I think very few (pure) math majors had any issue with calc.

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

#177

Post by optionalparens » Thu, Oct 20, 2016, 5:22 AM UTC

kenjackson wrote: post #134

optionalparens wrote: post #123

Earlier quoted context omitted.

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.

Oh I agree with that notion, it's just hard to get a lot out of stats without a large volume of it and a lot of other types of math as tools. Some are pre-requisites, while others are simply useful for applying in conjunction/situationally.

I still feel like calculus is one of the more valuable types of math. It really does teach you a lot, but like anything, it requires a good teacher. I find it all ties together for me in the end, like when doing graphics programming and games, I use a lot of trig, geometry, algebra, and calculus even, and stats some (and even more for business-type stuff).

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

#178

Post by zumu » Thu, Oct 20, 2016, 5:48 AM UTC

castratikron wrote: post #10

AP Calculus was the most useful course I took in high school. It was rigorous, and I scored high enough on the exam to get two free college classes and start freshman year at the sophomore level, which left room at the end of my college career for more specialized courses. There should be some vetting process for the students who want to take AP classes. A lot of the students in my AP Calculus class had no business b…

This sounds like a defense of AP classes and not calculus in particular. Mind you there are also other AP math courses, notably statistics.

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

#179

Post by tnecniv » Thu, Oct 20, 2016, 6:09 AM UTC

I have a specific problems with the HS math curriculum I experienced (as opposed to the general problems with the whole math curriculum I had before my junior year of college). Mainly, there was just so much wasted time it was unbelievable.

I actually cannot remember a single thing that I learned in Algebra II because I either relearned it in Pre-Calc or never used it again. Similarly, we spent the whole first semester of Calc BC redoing all of Calc AB. What a waste!

If I could, I would replace all that wasted time with Linear Algebra (with proofs) and Probability (throw in statistics if you want). Calculus should stay, but teachers should motivate it better (this goes back to the general problems I alluded to). Those two classes are immensely useful and teach a whole different modes of thought compared to the regular pre-college curriculum.

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

#180

Post by elihu » Thu, Oct 20, 2016, 6:11 AM UTC

analog31 wrote: post #120

elihu wrote: post #106

Earlier quoted context omitted.

For a lot of people, geometry is their primary or perhaps only exposure to proofs. Not that everyone needs to understand proofs in their day-to-day lives, but it's a good mental discipline to at least be exposed to, and geometry proofs are fairly straightforward and relate to things that many people will have a good intuition to convince them that the thing proved by each correctly-constructed proof is really true. A…

From what I can tell, proofs are gone. I got my daughter all excited about geometry, by promising her that she could do proofs. The curriculum was almost exclusively focused on problem solving based on remembering formulas. She went ahead and did all of the proofs anyway.

:(

I hope that's not universally true, and that proofs are still actively taught in some high schools.