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Do the math: too much calculus? (2012)

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

#201

Post by disgruntledphd2 »

retsibsi wrote: post #174
montyhallstuff wrote: post #165

Earlier quoted context omitted.

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.

Indeed. This is the major reason I don't like the Monty Hall problem. When using stats and probability (at least in science), the universe is not your adversary.

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

#202

Post by dredmorbius »

lacampbell wrote: post #164
dredmorbius wrote: post #143

Earlier quoted context omitted.

Interesting. I've been thinking about education and a few dynamics of it, including human cognitive development and individual differences. 1. Young children seem to have the biggest capability in learning languages. 2. There's the John Stuart Mill example -- homeschooled by his father in the classics and philosophy. Interesting biography (Wikipedia has a fair account). 3. There are children who are naturally skilled…

> 1. Young children seem to have the biggest capability in learning languages. Is there any evidence that this is actually due to their capability? I would say it's far more likely it's their environment. IE, people are constantly talking to young kids. They simplify what they say to make themselves understood. And children have a huge incentive to learn to communicate - to get what they want from parents, or just un…

There's extensive research literature strongly suggesting an age link: https://scholar.google.com/scholar?q=language%20learning%20a...

I'm also aware of a few other age/skill type relations.

Musical ability frequently manifests early -- by age 4-5 in numerous historical cases.

Talents requiring physical coordination -- ballet and gymnastics -- around age 9-10 by Russian tradition.

Maths and scientific curiosity perhaps early adolescence.

Complex logic and activities requiring a fully developed inhibitory function (avoiding reflex and rash actions) possibly early 20s.

I'm suspecting other forms of more synthetic (as in: assembling from parts) logic might not develop fully until the 30s or 40s. Other skills might similarly not develop until later.

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

#203

Post by dredmorbius »

abecedarius wrote: post #160
dredmorbius wrote: post #143

Earlier quoted context omitted.

Interesting. I've been thinking about education and a few dynamics of it, including human cognitive development and individual differences. 1. Young children seem to have the biggest capability in learning languages. 2. There's the John Stuart Mill example -- homeschooled by his father in the classics and philosophy. Interesting biography (Wikipedia has a fair account). 3. There are children who are naturally skilled…

My guess is that math teaching can be a benefit to anyone at any age, but the usual curriculum on average mainly teaches people to hate math. (That's what's happening with my niece in grade school right now.) For some kids arithmetic may be boring, but at least it's easy, and they suffer less of this damage. If you start it later when they're readier, more of them will be open to learning a little actual math, not ju…

I've been looking at John Cleese's presentations on creativity, and suspect he's very much on to something with the idea of play and humour as key to real learning. I also think that there are approaches which take this too far (video games and such), though I've learned a number of things from unexpected places.

Neal Stephenson's Cryptonomicon, The Baroque Cycle, and Anathem, for example, are all explorations of ideas. The first two of business, banking, and money, in contemporary and historical times, respectively. The latter covers a great deal of historical thought.

Better authors of topics often incorporate humour in their writing -- someting the past few generations of academic literature would do well to consider. OTOH, it's a cure for insomnia.

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

#204

Post by fao_ »

fao_ wrote: post #64
baldfat wrote: post #56

Earlier quoted context omitted.

The beauty of calculus is that you actually understand the Real World. For example I taught my daughter factions this way. Me: "These factions you see them? Now see these decimal points?" Her: "Yeah I hate factions they are stupid!" Me: "Factions are the only consistently real numbers below 1 and decimal points for the most part are made up numbers. 1/3 is 1/3 and never .3333." Her: "I always thought it was the otehr…

... Ohhhhhh. I wish you would teach me calculus =)

This... might have come out wrong. What I meant was something along the lines of: You should teach, have you considered mentoring?

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

#205

Post by pdm55 »

douche wrote: post #102
jimmydddd wrote: post #13

For lower level math, my kids seemed to revisit the same subjects over and over from k-9th grades. For example, mean, median and mode, year after year. The first few years it was just robotic problem solving. After a few years, they started to get a broader understanding. Maybe some people need to take several passes through the higher level stuff too.I guess the problem is that there is not enough time.

What infuriates me, looking back on it, was simply how much of my time the educational system was allowed to waste. Thousands and thousands of lost hours, some of them filled with pointless busy-work, but many more just squandered on inanity. At least some of the teachers would let you finish the daily penance quickly, and sit quietly in the corner reading something interesting. Too few, though.

I look at this way: early education focuses on learning to read and write and do arithmetic. The basic purpose is to give you tools whereby they can continue to educate yourself throughout your life. So basically, our early education aims to put our future education into our own hands.

Let me qualify the above a little. You might ask, well what are the purpose of high-school teachers or university lecturers? Once we are given these tools shouldn't we just be given the books to progress on our own? Hopefully, teachers are people who can take a complex subject and guide us towards an understanding of that subject. To me teachers are like guides through a jungle of knowledge. We need a guide so that we don't get lost. For me, a teacher is part guide and part "composer of problems". I consider teaching to be guided problem-solving. When I am teaching a course, I pose problems, whether it be in Math or Chemistry, and guide the students as they try to solve those problems.

And as for your comment about "wasted time", I am not sure what to think about this. The difficulty with learning is that we need repetition and reinforcement. Yes, I see a lot of pointless repetition, but we do need some repetition. I am presently struggling with improving my chess skills, as well as trying to learn to program an Arduino. Mastering these skills require a lot of repetition in order to cement that knowledge. Wish me luck.

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

#206

Post by pdm55 »

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.

Have you tried Geogebra?

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

#207

Post by thearn4 »

eli_gottlieb wrote: post #153
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 with trying to teach statistics, to anyone really, is that once you get past the very basics, you have to stop every time the lecturer/textbook invokes measure theory and figure out how to explain the concepts without just making everyone sit through two to three semesters of real analysis.

Agreed, there is definitely a huge gap between doing, for example, a t-test and deriving the t-test (or giving any substantial justification for its form). You need a class in analysis, a class in measure-theoretic probability, then a mathematical statistics class to bring you back to the topic. It's a tough square to circle.

Most who teach stats at the undergrad or hs level have little choice but to defer a lot of things to a later course.

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

#208

Post by BigJeffeRonaldo »

IndianAstronaut wrote: post #168
analog31 wrote: post #144

Earlier quoted context omitted.

For students who aren't going to learn calculus, I wonder if a better introduction to prob and stats (and perhaps math in general) would be through simulation and visualization rather than learning formulas by rote. I had the opposite problem, which was that I learned stats as a bunch of proofs with practically no applications. I only use fairly basic stats today, but I often test my conclusions by running some sort…

>practically no applications This is the general problem with math education in general. Just equations. Solving equations is relatively easy. Coming up with them is where the real challenge and innovation is.

Solving equations is NOT easy.

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

#209

Post by BigJeffeRonaldo »

baldfat wrote: post #56
CalChris wrote: post #9

Earlier quoted context omitted.

Berkeley EECS grad. I didn't go the device physics route but I used calculus about twice in 4 years except in physics classes.

The beauty of calculus is that you actually understand the Real World. For example I taught my daughter factions this way. Me: "These factions you see them? Now see these decimal points?" Her: "Yeah I hate factions they are stupid!" Me: "Factions are the only consistently real numbers below 1 and decimal points for the most part are made up numbers. 1/3 is 1/3 and never .3333." Her: "I always thought it was the otehr…

https://www.dpmms.cam.ac.uk/~wtg10/decimals.html

You can construct a complete ordered field from infinite decimals

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

#210

Post by danielmorozoff »

TheOtherHobbes wrote: post #197
danielmorozoff wrote: post #131

Earlier quoted context omitted.

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…

In the UK high school math is really a preparation for a hard science/engineering degree, and there's a lot more calc than stats in hard science/engineering. That's why calc is central. I wouldn't be surprised if it's the same in the US. This means most people have no idea what the math they learn is used for, because they never reach a level where they have to use it. So they leave school with the idea that math is…

Having taken hard science in university/ during my PhD, it's hard for me to say that calc presents itself more than stats. Perhaps in engineering but only in the early years of uni, physics same thing- year two is an exposition to quantum mechanics... In chemistry and biology I would say stats/prob definitely predominates. In chemistry calc begins to appears in physical chem.

Why most people don't understand the math they learned and what it is used for, I would argue has much more to do with how they are not taught to interpret what they see around them mathematically- granted that takes years of practice, but we do not care to teach this to children.

Now streamlining math is all well and good, but we may loose some of the greatest mathematical minds by doing so. Many famous mathematicians (Euler, Lagrange for example) were slated to be someone else- professionally that is. And it took other great mathematicians to notice them.

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