Should Probability and Basic Stats Replace Calculus in the High School Curriculum?
Don't get me wrong, I love calculus. I use calculus every day. If you're a bright young student with dreams of becoming a mathematician, physicist, engineer, or economist, a good foundation in calculus is just what you need to get started. For everyone who isn't that kid, calculus may be something they learn once and then never use again.
Probability, on the other hand, is something that we all deal with all the time, and it's something that humans have a remarkably hard time grasping intuitively.
Take the famous Monty Hall problem, for instance, which fools even very smart people:
People don't understand this problem because they use unconditional probabilities when they should be using conditional probabilities. People make other mistakes, too, like the prosecutor's fallacy, which is another confusion about conditional probability.
Here's an example of how conditional probabilities come into play in everyday life (I've made up the numbers). Say you're a 60-year-old male and you'd like to plan for your retirement. You know that 90% of males in your country have died by the age of 85, so you figure if you have enough assets to keep paying your living expenses until age 85, there's only a 10% chance you will run out of money before you feel death's sweet embrace. Wrong! You've actually got a better than 10% chance of surviving past 85 because that 90% statistic includes all the people who didn't make it to 60. The relevant statistic is the number of people who made it past 85 given that they had made it to 60. A high school class on basic probability could really help people plan their lives.
The other reason I would put probability and statistics in the curriculum is to help people not to be fooled by the kinds of misleading statistics thrown around by politicians, journalists, and careless or malicious academics. In the latest episode of Jim Pethokoukis' podcast, he and Richard Burkhauser discussed the misleading data from a study by two French economists, Piketty and Saez, whose statistical analysis seemed to indicate that middle-class incomes had remained stagnant over the past twenty years while the richest 5% had gotten dramatically wealthier. Burkhauser showed that Piketty and Saez' results were very sensitive to their assumptions, and that with better data and better assumptions median income growth was not 3% over 20 years as Piketty and Saez claimed, it was more like 36%, and inequality isn't growing out of control but has remained fairly constant since 1992.
Believe it or not, I saw the graph from Piketty and Saez' study pasted to a sign at my local Occupy rally earlier this year (I was a passerby, not a participant). There are people in this world who went out and slept in tents and were a general nuisance to everyone on the basis of bad data. A little bit of statistical knowledge might not be enough to have every single Occupy person debunk a study that most economists failed to see through, but it would at least teach them that statistics are often misleading and shouldn't be taken unquestioningly as facts.
If I were designing a high school statistics course, I wouldn't just transfer "Intro to Statistics" from university to grade 12 (I was a TA for "Intro to Statistics" this past term; it's frightfully boring). When you teach someone to drive, you show him the brake before you show him the gas pedal. When you teach someone to use a gun you tell him not to point it at his face before you show him how to take the safety off. In statistics we do the opposite. Sure, there's the obligatory speech about how correlation is not causation, but that is quickly forgotten in the rush to show the students all the amazing things they can do with statistics. It's not until grad school that students are taught how to really think critically about statistics and to question the assumptions of statistical models.
However, you don't need four years of stats courses to start questioning assumptions. Thomas Sowell, in his many books, critiques bad statistics and the inferences drawn from them in terms that anyone can understand. There are some statistical pitfalls that really do require years of statistical training to understand, but people should at least know that they exist and that statistical "facts" are few. What's key is for people to develop a healthy scepticism towards statistics, so they can't be easily manipulated by politicians and intellectuals who may not have their best interests at heart.
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Jul '10
Re: Should Probability and Basic Stats Replace Calculus in the High School Curriculum?
Teach ethics and virtue. You can get to statistics later.
Dec '11
Re: Should Probability and Basic Stats Replace Calculus in the High School Curriculum?
I would hope they'd have time for ethics, virtue, and statistics. The Occupy Wall Street people could use a healthy dose of all three.
Jul '11
Re: Should Probability and Basic Stats Replace Calculus in the High School Curriculum?
I like the idea. I have had a running argument about the Monty Hall problem with a Stanford computer geek for years. Since it is on You Tube I am finally going to slam that geek in to submission.
Feb '11
Re: Should Probability and Basic Stats Replace Calculus in the High School Curriculum?
There is a pretty good argument that we should drop a lot of subjects from the high school curriculum. It seems I read a professor of some science subject advocating exactly that recently. Why waste time on subjects that are only stepping stones to careers you aren't likely to pursue? I wish I could find a reference to that article. I'll have to go looking.
Nov '10
Re: Should Probability and Basic Stats Replace Calculus in the High School Curriculum?
If it weren't for the fact that Paul Erdos, father of the probabilistic method in pure mathematics, confessed difficulty grasping the explanation for the solution to the Monty Hall Problem, then I would say your assertion that it "fools even very smart people" is nonsense, because as far as I'm concerned, the explanation is completely transparent. Erdos was unquestionably one of the smartest mathematical minds of the 20th century. I suspect what he meant was more along the lines that it violated intuition, and he was not prepared to entirely abandon that intuiton.
Well, if you like this problem so much, I have one that I consider superior to it, and it is every bit as elementary:
Two apparently indistinguishable envelopes contain perfectly negotiable blank cheques for large sums of money. Nothing is known about the actual amounts, except that it is known that one of the envelopes contains exactly twice as much money as the other.
You open one envelope. It contains a cheque for $1000. You are allowed to keep the $1000 or to switch to the other envelope. Which should you do, in order to maximize your expected profit from the venture?
Feb '11
Re: Should Probability and Basic Stats Replace Calculus in the High School Curriculum?
This isn't the one that I was thinking of, but it is along the same lines, and advocates a similar sort of stats course.
Nov '10
Re: Should Probability and Basic Stats Replace Calculus in the High School Curriculum?
To be clear, you now know that the other envelope contains either $500 or $2000. Perhaps it is obvious that there is a .5 probability of each, and also obviously, the two possibilities are mutually exclusive, comprising a complete set of possible outcomes. Thus, by elementary probability theory, if you switch to the other envelope your expected take is (.5)x2000 + (.5)x 500 = $1250, so by switching you could expect to enrich yourself, on average, by an extra $250.
Now wait a minute, your evil twin argues -- this argument will work no matter how much money is found in the first envelope. Therefore it must always be true that switching is the better option for maximizing one's profit. Therefore, the moment you decide which envelope you want, why not just open the other one and walk away?
But, you now wonder, how would that be distinguishable, in practical terms, from deciding to select that envelope in the first place? And is switching back even better?
Clearly our analysis has gone astray. Where? And there still remains the original question ... stay, or switch?
One last clue: Despite the superficial similarity, this problem is not isomorphic to the MHP.
Edited on December 8, 2012 at 7:10amNov '10
Re: Should Probability and Basic Stats Replace Calculus in the High School Curriculum?
Precisely the sort of person who should have no problem with the correct answer. In my work (math professor) I have met many people who have trouble with this. Some people can't be convinced by rational arguments. However, I have seen numerous computer scientists won over by seeing the evidence for themselves: They write a very simple code to model the game and run it, oh, say, a million times. Swapping wins the car 2/3 of the time; staying wins it only 1/3 of the time; they're convinced. This has led me to believe that, while such people may be perfectly rational, they are not particularly logical or analytical.
Sep '10
Re: Should Probability and Basic Stats Replace Calculus in the High School Curriculum?
As others have already observed, many mathematically-trained people get the Monte Hall problem wrong. Both calculus and statistics are important for someone in a scientific field. Calculus is more immediately useful, and indeed, prerequisite for any college-level physics. Statistics can wait for that purpose. As for the rest of the folks, most will take neither calculus nor stats, so it's a moot point. More specifically, the average juror (viz. prosecutor problem) will not take stats anyway so it won't help. The same goes for all similar issues in public policy (e.g. the cost/benefit of vaccines).
The Occupy Wall Street/Piketty and Saez issue is different. The OWS people like the misinterpretation given to the Piketty and Saez data. This is more a matter of confirmation bias than statistics. Perhaps a class in logic would help.
Nov '10
Re: Should Probability and Basic Stats Replace Calculus in the High School Curriculum?
Math is my profession, and I have a different take on the issue of why one ought to learn math, and the corollary issue of which math to prioritize.
I don't think the primary issue is whether or not a subject is likely to be "useful" for you later in life. Math (and literacy/language) is what separates us from the beasts. It is the quintessential human skill. That math is useful in a myriad other, practical ways, is mere economic gravy. Hardly anyone thinks of math in this way, but I am convinced that is where its main value lies. But math is so vast we must choose. The choices that have stood the test of time are for those subfields with the most connections to other things.
Edited on December 8, 2012 at 9:48amSep '10
Re: Should Probability and Basic Stats Replace Calculus in the High School Curriculum?
R. Craigen
I don't think the primary issue is whether or not a subject is likely to be "useful" for you later in life. Math (and literacy/language) is what separates us from the beasts. It is the quintessential human skill.
I don't dispute your point regarding math literacy in general. However, most students never reach calculus. I'd be happy if even a lower mathematical literacy were broadly attained. That would be a step up for most students.
The question originally posed is about calculus. Many of the students who take this class do have a very real use for it, at least as prerequisites for other classes they need. My point was simply that calculus is more valuable, perhaps even essential, for its target audience.
The cultural objective of mathematics education is addressed at a lower level, at least for now. While I would celebrate if most kids learned calculus, it's not going to happen soon. I remind you that there was a thread a few months back in which it was suggested that algebra was too much math for students. In that context, we're fortunate that kids even attain that level.
Sep '12
Re: Should Probability and Basic Stats Replace Calculus in the High School Curriculum?
Facts don't matter much in politics because their authority is low in the hierarchy of authorities. Like religion, politics' highest authority is belief. And like most conservatives who disputed Nate Silver on the belief that Romney was going to win (he had to!), we all learned that belief endures contrary facts.
Indeed, belief-source authority is the most robust and enduring kind of authority, stronger even than facts and logic tools like calculus, probability, and statistics. Why? Many reasons for this, but the most obvious one is that beliefs operate at a meta-level above other authority sources (such as sensory perception) in a way that organizes, makes sense of, and lends purpose to their products.
I heard somewhere that when communism was unraveling in Poland, the following graffito become common: 2+2=4 What did this graffito really mean? It meant "We no longer believe in communism". Back when belief in communism was robust, no facts, reason or logic could refute it. Only when belief faded did facts, reason or logic reassert their authority.
So no, teaching kids logic won't put paid to stupid beliefs.
Edited on December 8, 2012 at 12:04pmDec '12
Re: Should Probability and Basic Stats Replace Calculus in the High School Curriculum?
No choose. Teach both. Math good. Sports history bad.
Oct '12
Re: Should Probability and Basic Stats Replace Calculus in the High School Curriculum?
There is a 95% chance you are right. Stats are more important than calculus. I have a Bachelor's of Arts, Master's of Science, and Medical Degree. I taught statistics at the college level as an instructor, never have had a calculus class. It worked for me.
May '10
Re: Should Probability and Basic Stats Replace Calculus in the High School Curriculum?
I vote for stats over calculus.
Nov '10
Re: Should Probability and Basic Stats Replace Calculus in the High School Curriculum?
No takers for my alternative to the MHP?
Dec '12
Re: Should Probability and Basic Stats Replace Calculus in the High School Curriculum?
There are no conditional probabilities in your example. Odds are 50% you'll get richer and 50% you'll get poorer, no matter which envelope you pick first. It's a coin toss.
Aug '12
Re: Should Probability and Basic Stats Replace Calculus in the High School Curriculum?
Both are important fields. Calculus teaches you how to think about really big numbers and really small numbers and how math can be used to understand many parts of the world. It is the abstract way of dealing with reality, and this is good skill to have -even if you never do another derivative again.
Probability teaches you to consider patterns logically rather than to accept that what you see must result from the obvious pattern. This is another skill which is good to have -even if you never calculate another mean again.
Ideally, we'd learn both in High School.
As for Monty Hall, I don't know that the problem is statistical rather than verbal. Not everybody thinks Monty knows where the car is right away. If Monty is just opening a random door, he's not actually giving you information for a conditional probability.
Dec '11
Re: Should Probability and Basic Stats Replace Calculus in the High School Curriculum?
I've been working on it. If x is the smaller amount, then switching gives an expected value of (1/2)x+(1/2)2x=(3/2)x, and staying gives the same expected value. Knowing one of the values shouldn't make any difference unless you know more about the function from which x was drawn.
Nov '10
Re: Should Probability and Basic Stats Replace Calculus in the High School Curriculum?
Garrett Petersen
I've been working on it. If x is the smaller amount, then switching gives an expected value of (1/2)x+(1/2)2x=(3/2)x, and staying gives the same expected value. Knowing one of the values shouldn't make any difference unless you know more about the function from which x was drawn. · 3 minutes ago
And if x is the larger amount switching gives an expected value of (3/4)x. Now there is a 50% probability of each ...
:-)