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Do the math: too much calculus? (2012)

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

#161

Post by abecedarius »

asimeqi wrote: post #101
abecedarius wrote: post #82

Earlier quoted context omitted.

A U.S. elementary school principal in the 20th C. got his school to try leaving out any math in the first three grades (iirc), and the kids reportedly came out fine. I can't remember his name, and I'd like to be reminded too.

It was L. P. Benezet, superintendent of schools in Manchester, New Hampshire, who experimented with teaching no math in the first five grades in the early 1930s. See for example https://www.psychologytoday.com/blog/freedom-learn/201003/wh...

Thanks! There's also his own report: http://www.inference.phy.cam.ac.uk/sanjoy/benezet/1.html

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

#162

Post by thelambentonion »

dguest wrote: post #125
rconti wrote: post #84

Earlier quoted context omitted.

IME the #1 problem is the failure to teach anything WELL. I took AP Calculus in high school, and today, as a 35-year-old, I'm still continually surprising myself with the problems I realize must be related to Calculus-- but nobody bothered to tell me. Things like the 'ol speed vs acceleration vs jerk kind of problems. All calculus ever taught me was to apply these formulas to those problems (for some reason) and that…

When calculus is taught badly no one knows how to apply it even when the problems are staring them in the face. I TA'ed physics at one of the top Ivy League universities in the US, and what I found amazing was that we didn't require calculus for our introductory mechanics course. You don't get into these universities without taking all the highest level courses in high-school, which means that almost everyone in the…

> Math should be motivated by science and modeling, otherwise it's just an exercise in diddling numbers.

I like a lot of what you said, but I do disagree with that point. Immediate application of mathematical concepts may be gratifying, but making it the _only_ motivating factor behind maths can result in students who conflate the underlying machinery with its application [1].

Taking a pure maths course in undergrad with related subject matter prior to one of my Controls classes made the class substantially easier to understand. Learning the abstract concepts beforehand let me see the common applications more quickly than others.

I think the biggest difference is that when I learned some Calculus through Physics, it was a little more difficult to go from a concrete basis to a more general one. My understanding ended up being based on analogies to other concepts until I went back and covered the theory again.

[1] This is all my own opinion, I'm not an educator so the most I can do is pull from my own pedagogical experience :)

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

#163

Post by krschultz »

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.

That's a good point. I'm not an educator and it's been a while so I can't recall the sequence of stats 101, calc 1, more stats, more calc, etc. It all blurs together at some point.

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

#164

Post by lacampbell »

dredmorbius wrote: post #143
abecedarius wrote: post #82

Earlier quoted context omitted.

A U.S. elementary school principal in the 20th C. got his school to try leaving out any math in the first three grades (iirc), and the kids reportedly came out fine. I can't remember his name, and I'd like to be reminded too.

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 understand the world around them. The same isn't true for a grown adult trying to learn a second language, in isolation, that they don't really need in their day to day lives.

Disclaimer: this is pure conjecture.

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

#165

Post by montyhallstuff »

posterboy wrote: post #159
russellspitzer wrote: post #151

Earlier quoted context omitted.

I prefer thinking about the 100 doors case. You are shown 100 doors 1 ... 100. Imagine you pick 1 door. Of the other 99 doors Monty tells you which 98 of them don't have the prize. He now asks if you want to switch to the 99th door that he didn't cancel out. Intuitively that seems to make much more sense to me.

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 that the chosen card is a black face card? E = {J-club,Q-club,K-club,J-spade,Q-spade,K-spade}. What is the probability that the chosen card is a black face card? 6/52 by the formula above.

Monty Hall problem can be solved by using the above as a guide.

Sample space here : {switch, switch, stay-with-your-original-choice}. By the formula above, the probability of switching is 2/3 and the probability of staying put is 1/3.

If you're doubting the formula, I can provide you with a proof.

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

#166

Post by eli_gottlieb »

hilop wrote: post #158
eli_gottlieb wrote: post #153

Earlier quoted context omitted.

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.

You just use discrete scenarios, not continuous. Same concepts, easier math. same way you can do linear geometry without calculus.

In my experience, just knowing discrete probability and statistics is profoundly unuseful, especially when you walk out of class thinking you know the subject, sign up for a next course that involves real-valued quantities, and bomb everything.

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

#167

Post by [deleted] »

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,…

[deleted]

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

#168

Post by IndianAstronaut »

analog31 wrote: post #144
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…

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.

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

#169

Post by ivansavz »

yequalsx wrote: post #78
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,…

I think that a student not well versed in how to perform calculations will not do well with the theory. Last night in the calculus class I am currently teaching a student could not correctly translate the fact that f is increasing where f' > 0. This on a problem where f' was linear. He just stared blankly at the inequality 2x-3>0. Anyone well versed in this sort of calculation is not going to fumble around at this pr…

I'd be very interested to hear your opinion about a book I've been working on. It's a 3-in-1 combo, math+mech+calc, I wanted to teach mechanics and calculus in an integrated manner and I realize, from 15 years of private tutoring, I need to review all the high school math material because many students don't know the basic.

Please drop me a line if this sounds interesting. Email in profile.

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

#170

Post by ivansavz »

japhyr wrote: post #41

I have a 5 year old, and I was talking to some acquaintances who also have a kindergartener. The kids are learning math, by playing with manipulatives. It's great; they're learning mathematical concepts, and they have no idea anyone in the world doesn't like "math". They don't know what "math" is, they are just starting to discover that the world is made up of interesting but identifiable shapes, and that numbers hav…

I don't have kids, but if I had a 5 year old, I'd sure get them some of these: https://en.wikipedia.org/wiki/Cuisenaire_rods

I've often thought about inventing math toys, e.g., group theory for toddler: S_3 = triangle toy, S_4 square toy, etc. I'm sure there are many advanced math topics that could be turned into toys. No symbols, no equations, just toys, but subtly you're getting kids familiar with important math structures.

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