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've rarely used calculus outside an academic setting and I use at least basic statistics/probability thinking all the time.
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.
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 annoying bullshit that they don't understand and can't use.
I think it would be better to stream math by ability much earlier. Make everyday math compulsory for everyone - interest rates and loans, basic stats, currency rates, areas/volumes, simple trig, maybe some slightly more advanced finance, perhaps touch on some calc ideas.
Only teach calc and scientific math to those with the talent and interest for it. Give everyone else a taste so they understand a little, but don't try and get them ready for a career they'll never go into.
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.
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.
Right before any of the doors is opened we have no idea what to expect, so there are 3 equaly likely possibilities:
1. prize is behind A 2. prize is behind B 3. prize is behind C.
In case (1) you win by staying with your choice of A. In case (2) Monty Hall must open C (by the rules), so you'll win if you switch to B and in (3) the host will open B(by the rules) and you should switch to C.
In 2 cases out of 3 (2/3) you win by switching and in only one case out of 3 (1/3) you win by staying with your first choice. Since 2/3 > 1/3, you should switch.
This problem is easy if you know the defintions of sample space, event and the Equally Likely Probability Formula.
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.
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 is much more apparent the smaller the initial chance is. But I only intuitively used it to defend my position.
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.
Two thirds of the time, you pick the wrong door on your first choice. So far, so obvious. Now, you picked a wrong door, Monty opens the other wrong door, and there are only three doors. Obviously the remaining door has the prize.
Every time you pick the wrong door, and then switch, you win. That means two thirds of time, if you switch, you win. That makes switching the right strategy.
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 flying is safer than driving, you need to know how to compare. A lot of stats thinking is also not formulas, but qualitative things like "Am I comparing apples to oranges".
Calculus by contrast is pretty common sensical. Acceleration is the derivative of velocity, you find the local extremes by finding the zeros of the derivative, the integral of a net shaped field is the rim... that all makes sense with some pictures.
I also haven't come across it much in finance, apart from derivatives work, where funnily enough you can't do with high school calculus, but you need stochastic calculus. Which contains some quite statistics related concepts.