Teaching statistics before calculus?

[Greg Bernhardt]

Here is a short (and old) TED talk where a mathematics professor suggests we teach stats and probability in depth before teaching Calculus because it's math that is more relevant to a wider range of people. Have we got our math curriculum wrong? Thoughts?

 

[axmls]

I absolutely would love if statistics was taught extensively to every student. In this day and age, where half the news stories out there involve some kind of statistic, it's simply necessary to have a population that understands statistics, sort of like Sagan's "living in an increasingly technological world where fewer people understand technology."

Unfortunately, statistics isn't often taught well. Regardless, it's so important that people be taught when to recognize bad statistics or when it's clear statistics are being used to stretch the truth.

 

[Diaz Lilahk]

Here is a short (and old) TED talk where a mathematics professor suggests we teach stats and probability in depth before teaching Calculus because it's math that is more relevant to a wider range of people. Have we got our math curriculum wrong? Thoughts?

The fundamental issue of mathematics teaching is not the order with which the content is taught but the content itself. By and large the content that is taught is the routine manipulation of ready made functions, regardless of whether it is algebra, geometry, trigonometry, calculus or statistics. The reason that this is the case is that this is the only mathematics which is politically acceptable from the standpoint of our high schools being transcript/diploma mills for entry into college, which is itself a gatekeeper to the corporate workplace. If deeper thought was taught and assessed it leads to outright rebellion on the part of the major stakeholders of the public k-12 enterprise, namely the students and parents. Before long, pressure is exerted on administrators who then transfer it to teachers.

Consider this, our high school Algebra/Pre-calculus courses should be capped off with an extensive treatment of the theory of interest. It is fairly comprehensive in terms of utilizing the mechanics of Algebra and it is even more important to our everyday lives than statistics, so why is it given only the most superficial treatment? It is too difficult and requires a algebraic competency which is beyond the abilities of the vast majority of our students precisely for the reasons stated above.

 

[late347]

Finnish high school curriculum teaches probability and statistics in course, which is situated as a pre-calculus course.

courses for higher math in high school
1.functions and equations
2. polynomial functions
3, geometry
4. analytic geometry
5. vectors
6. probability and statistics
7. calculus I (mainly derivatives, limits)
8. root- and logarithm functions
9. trigonometric functions, and sequences and series
10. integrals

optional courses (advanced courses; if I recall correctly at least one is recommended such as: the eleventh or the twelfth or the thirteenth)
11. number theory, and logic
12. advanced numeric and algebraic methods
13. calculus II (advanced calculus course)
14. rehearsal course

 

[late347]

for the lesser advanced mathematics curriculum in high schools there is also probability and statistics being taught (as far as I know)

1.) Finnish high schools have two math pipelines generally speaking. Regular and advanced mathematics. (unless you are , in a private high school, in which case the curriculum may differ, I think)

2.) Or if you were homeschooled at high school... but homeschooling is more rare in Finland than USA. Kids are allowed to take the high school finals after home schooling, after all.

3. Or if you already chose trade school instead of high school. Trade schools have different curriculum compared to highs schools I think.

 

[chiro]

Teaching statistics properly requires quite some training and the only thing that can be taught is a very superficial view of statistics.

If people are going to interpret statistics properly then they will need a lot more than a one or two semester course on it.

 

[Andy Resnick]

Here is a short (and old) TED talk where a mathematics professor suggests we teach stats and probability in depth before teaching Calculus because it's math that is more relevant to a wider range of people. Have we got our math curriculum wrong? Thoughts?

I've heard of this before, and discussed it with my Math colleagues. There is merit to teaching more/better statistics in High School, no question.

 

[gleem]

Statistical thinking will one day be as necessary a qualification for efficient citizenship as the ability to read and write.
--H.G. Wells

Teaching statistics properly requires quite some training and the only thing that can be taught is a very superficial view of statistics.

If people are going to interpret statistics properly then they will need a lot more than a one or two semester course on it.

Quite so.

Clearly a couple of semester courses in statistics does not make a person a statistician. But at some point as has been previously noted people today must have a basic knowledge of the concepts and methods of statistical analysis. They must appreciate the misuses of statistics , the error in interpretation of data, that correlation does not imply cause and effect and the down right use of statistics to mislead. It is not just about definitions , procedures and formulas. I do not think that a good elementary statistics course with the aim of adequately preparing a person to appreciate the value and danger of statistical inference can be taught by just any math teacher.

 

[Mark44]

In this day and age, where half the news stories out there involve some kind of statistic, it's simply necessary to have a population that understands statistics, sort of like Sagan's "living in an increasingly technological world where fewer people understand technology."

59.3% of all statistics are made up on the spot...

 

[chiro]

I see what you are saying gleem but the thing missing is context.

One can't really have that context without going a bit further and when you have a superficial form of a logic then regardless of how good that logic is it will be interpreted superficially as well.

Take correlation can be different from causation as an example.

The thing of this has more to do with how much variation (and co-variation) exists between the processes and this involves more than just co-variance and correlation.

If people learned this and then said "Well the rule says this so don't believe it" then you have people parroting something they don't understand and using it in ways they don't really comprehend and (which is worse of all) making inferences that they aren't really in a position to make.

One of the worst things you can do for people is give them a superficial, non-comprehensive form of a logic and get them to think that it isn't superficial.

It's a dangerous thing for people to think they know something when they don't and then apply it to the world around them because they will have not enough doubt and way too much confidence and make assertions and inferences that may be out of line for what should be done.

For the above reason I'd rather people just admit they don't understand things as opposed to thinking they did because they took an introductory course and now suddenly they think they can make interpretations on logical systems that they themselves don't understand.

This post is not just for statistics but for any logic and it's a really bad idea to get someone thinking they are an expert when they really aren't in a position to even decide whether they are or not. To really know something takes a lot of work - and that is something small. Encouraging people to think they are in a position to make decisions and comprehend a situation they can't is a dangerous and outright irresponsible thing to do and I certainly would not encourage that myself.

 

[mfb]

For the above reason I'd rather people just admit they don't understand things

Yeah, but how often does that happen if the topic is not directly politics but with political implications?
Knowing that you don't know things needs some education as well. Also the knowledge that you can easily make up statistics that sound great until you look at the details.

 

[Andy Resnick]

<snip>Encouraging people to think they are in a position to make decisions and comprehend a situation they can't is a dangerous and outright irresponsible thing to do and I certainly would not encourage that myself.

That sounds nice and neat but is completely untenable. People go through their day continuously making decisions based on incomplete information, and even more, people engage in many activities of which they are not an expert. Is it irresponsible of me to have a garden, since I am not a master gardener? Of course not.

I suppose you would ban procreation...

 

[chiro]

I'm saying it has to be put in its proper context.

One of the worst things you can do for someones intellect is to get them to believe something they aren't.

Incomplete information always exists and that is a certainty - but that doesn't mean it's an excuse to just say superficial logic is ok.

Politics is a screwed up system anyway and causes so many problems when it comes to decision making so that alone is already bad enough as it is.

If you are gardener you don't want to apply pesticides and weed killer incorrectly since you might kill all of the healthy and wanted plants.

Same thing with statistics - you don't want people taking their new found knowledge and making inferences that would destroy far more truth than it creates.

 

[Andy Resnick]

One of the worst things you can do for someones intellect is to get them to believe something they aren't.
<snip>

Who am I to tell someone who they are or what they can (or can't) do?

Edit- let me elaborate a bit more, because now we are getting to the actual rationale for formal education: evaluation of student efforts by experts. Indeed, I do tell my students what they can or can't do within the context of my class.

Education should not have a 'value'. I realize this is contrary to societal trends during the past several decades, especially in higher education. Currently, a college degree is often treated as a commodity- something that is purchased by a customer, sold by the institution, and confers some sort of value- future job earnings, prestige, etc. When education is treated as a commodity- something that can be bought and sold and has value (monetary or otherwise)- institutions are transformed into service industries with all the concomitant negative associations- customer satisfaction requirements, administrative oversight of academic functions, accountability requirements, etc. Only by resisting the commodification of education can educators truly ensure that students have an opportunity to gain proficiency and master skills.

On one hand, the goal of introducing basic statistics earlier in the math curriculum can be phrased in terms of 'educational commodity' by producing smarter consumers- consumers of *information*. But fundamentally, providing formal instruction of statistics at any level should be the same as any other formal education exercise- student efforts are evaluated by experts. One may be tempted to argue that (for example) a junior-high school teacher is not a sufficient expert to provide proper evaluation, but that's clearly a false argument since with few exceptions, no primary school educator has a PhD degree (or even a MS degree in something other than education).

Educational curricula must be adaptable to accommodate changing needs. One current need is for the adult population to have appropriate intellectual tools to critically evaluate biased information- news feeds, press releases, etc. Schools are in a essential position to provide the necessary evaluations to *prevent* someone from growing up thinking they are more of an expert than they in fact are.

 

[axmls]

One of the worst things you can do for someones intellect is to get them to believe something they aren't.

Is this your view of education? Don't educate, because if you don't teach them enough to be experts, then you've done more harm than good?

Guess I shouldn't have taken any literature classes, lest I fall into the trap of thinking I can have opinions on books without years more of training.

The entire point is that in the modern world, people need to understand at least vaguely that not every statistic they read is the complete truth. They need to know the common techniques used to mislead people.

 

[chiro]

My value is to get people to differentiate the limits that an education can provide.

I'm all for education but when people take it out of context things can go very sour.

Identifying context (which is the main point here) is far more important to having that context retained and it's often lacking in people's minds based on the effort it actually takes to clarify it.

If you don't have that context you create people that think in a more absolute way and that contributes to poor logic, reasoning and inference.

As educated people surely you must understand this.

 

[MarneMath]

My only quam with this is that in my experience, most students end up taking a statistics course before they ever take a calculus course. I think the problem is that if you study STEM, then the inverse is true, so the professor has a view that that is the norm. I, on the other hand, know far more people who have taken Intro to stat methods or business stats or econometric stats, or some other name for stats before ever taking a calculus course (if they ever do take a calculus course).

While I understand completely that there is a portion of students whose end goal is to take calculus in high school, I don't think that ends up being the majority of high school students. I could be wrong though.

 

[chiro]

The calculus is definitely necessary for the calculation aspect but I think most people in this thread are debating the sort of conceptual offerings as opposed to doing the "nitty gritty" stuff you would need to make any sort of real thorough assessment.

If you are wondering how you could do it without calculus you could just get lots of statistical tables and get the student to use them rather than do the calculations with the calculus techniques - and that is exactly what a lot of introductory statistics courses do.

 

[brainpushups]

Today's SMBC reminded me of this thread

 

[gleem]

That may explain why I am such a poor consumer. So are we all (physicists/mathematicians) poor consumers?

 

[BiGyElLoWhAt]

Does not having money constitute being a poor consumer?

 

[mpresic]

I was educated in a NY State high school in the 1970s and we had concepts of probability mostly in our precalculus course after geometry. I enjoy probability but think of the unintended consequences, good and bad. Gambling is fascinating, but do we want students to know you should always split aces and eights when playing blackjack. We can give rise to great poker and dice players.

I am being somewhat facetious but at least the students (and adults) might find out the real meaning of the often misused term "double down". In blackjack, double down means you get one card and you either win or lose based on that one card. You have also put twice the money in the pot. Major political leaders and parties are said to double down when they are reemphasizing the negative attributes of their opposition, and they continue over and over again to "double down" (incorrect usage). If the party was really doubling down, they could only do it once, and their failure would lead to an immediate win by their opposition, This would certainly limit their ardor to double down. Could you imagine one candidate party gambling and losing two consecutive elections (immediately) rather than just one.

Aside over. Probability and statistics can get pretty deep and the professor discusses the move to digital (probability) over continuous (calculus) but probability and statistics treats continuous distributions, as well, for example the ubiquitous and important Gaussian distribution. Because some concepts are counterintuitive, the subject runs the risk of being "touchy feely"; more so than calculus.

I think probability would be fun to teach, and useful. Growing up I knew more adult family members and friends who went to Vegas, than Cape Canaveral/Kennedy. Most often adults tried to convince me to bet on red after black came down 5 times in a row. I reminded them (usually unconvincingly) the roulette wheel still contained the same number of slots for red as black for the next spin. (They wrestled with this puzzle)

By the way do we teach students the Monty Hall problem. We can start a lot of arguments.

I have to disagree about the politics involved in the choice of curriculum. I would like to see linear programming in the curriculum, rudiments of game theory in the curriculum, a little spherical geometry would be good, but the sad truth is there is just to much to teach high school students and choices have to be made. Compound interest is usually taught in the precalculus year when I went to school. Maybe it still is.

At least one does not have to teach, linear interpolation in trig and log tables, as was done when I went to school. Many other topics have been supplanted by the computer and pocket calculator. This frees up some time.

 

[gleem]

Today we have the statistics of polls. What can we teach our children with regard to this seamiling important statistic.

 

[andrewkirk]

Another vote in favour of teaching stats and probability here!

I'm not so concerned about whether it is taught before or after calculus, as long as it is taught! In countries like mine (Australia) where the vast majority of children finish high school, it is only necessary that it be covered before the end of high school. That could be before or after calculus. There are some advantages in doing at least basic calculus first, in order that the notion of expected value and the relationship between PDFs and CDFs can be properly understood.

Of course, in order to teach it, something needs to be dropped, to make way for it. That's easiest in advanced curricula. In my state (NSW), the highest level of high school mathematics has twice the number of contact hours that standard maths has, and includes complex numbers, conic sections, polynomials and advanced integration. Much as I love all those beautiful subjects, I concede that they are used by only a tiny minority of people in post-school life, whereas stats and prob would be of regular use. So it would be easy to swap stats and probability into the advanced maths curriculum.

For standard level maths it would be more difficult, because there are fewer topics that would be uncontroversial to cut. It would depend on what topics were in the curriculum. But I think most study of integration techniques could be dropped. All that's really needed for a high school graduate is to know that it's the reverse operation of differentiation and that it can be used to give areas under curves and other cumulative amounts. Learning all the fancy techniques can be left to university.

 

[bhobba]

Another vote in favour of teaching stats and probability here!

Its what they do in Australia and in the IB program.

It's taught in conjunction with calculus.

Thanks
Bill

 

[houlahound]

I think the norm is in statistics teaching...HAW HAW.

why is the calculation of the average taught to every student on the planet when it is the most misleading measure of central tendency and the median which is more useful typically only taught in electives.

Not sure how accurate that is but on average... Oh too much, cracking myself up.

 

[mfb]

The median can be misleading as well.
If I go purely by the median day, all days are without rain, I never go to the supermarket, and I work every day but never get paid.
Take an average day, and it rains 0.xx times, I go to the supermarket 0.xx times, I work on y% of the days and I get paid on average z Euros per day.

 

[brainpushups]

I'm reminded of Aaron Levinstein's famous quote "Statistics are like a bikini. What they reveal is suggestive, but what they conceal is vital."

If the argument is to teach probability and statistics so that more people can be better at making informed decisions then I think the efforts would be better spent on teaching critical thinking. A few well-constructed conceptual lessons on identifying pseudoscience and bogus claims can go a long way with students. I can imagine that, for students not interested in math or science, that lessons on critical thinking may even have more long-term benefit than an entire course in probability and statistics.

 

[axmls]

A few well-constructed conceptual lessons on identifying pseudoscience and bogus claims can go a long way with students.

This! I have had (high school) teachers that use pseudoscientific (read: completely false) sources when they teach! My school, at least, and many others, have never realized how important it is to teach students the art of healthy skepticism. One lesson on fact checking and skepticism is worth 100 lectures on anything science.

People forget that science isn't a list of facts. It's a way of thinking.

 

[Dale]

a mathematics professor suggests we teach stats and probability in depth before teaching Calculus

That is how it is done in my local public school system.

 

[newjerseyrunner]

I think statistics are more important to the everyday lives of the working class than calculus. Those of us who actually use calculus are few and far between, but everyone should understand the concept of normal distribution and bell curves. The fact that The Lottery is even a business shows a clear need for statistics education in the USA. Gambling is also getting more popular, especially casinos. I think this will be very bad for the society as a whole.

 

[houlahound]

The level of math required for life in the modern world for most people, in fact the vast majority is very basic.

Neither of my parents went to high school, my grandparents never went to any school. They all own or owned investment properties, ran businesses ... lived very well indeed.

My dear ol mother just this month designed a beautiful kitchen with a tape measure and computer and hit the submit button. Done it dozens of times. She buys and sells property, houses any shares. Does not know what calculus or algebra are. Only knows basic arithmetic.

it is a well built, well equipped modern kitchen you will find anywhere. Arrived on a truck in flat packs and was assembled easily and every measurement was perfect.

The hardest part was choosing the colours.

Educators IMO way overstate the level of math required for the life in today's society.

 

[bhobba]

People forget that science isn't a list of facts. It's a way of thinking.

That was Feynmans lament.

Students were often taught rote answers rather than how to think with it.

The IB program really tries hard to reverse that, but I don't know of others.

Thanks
Bill

 

[bhobba]

I think statistics are more important to the everyday lives of the working class than calculus.

True.

But like a Foreign Langauge the thinking skills it enhances is of value.

I like what the IB program does and what we do here in Australia - its taught together. We have a separate math stream for those that do not want to do calculus, but personally I would discourage students from taking it. Already there is concern where I am far too many people are taking the soft option. Universities have even had to change the way they teach stuff eg in economics a bit of calculus makes teaching some things easier but since students don't have it they have to go the less elegant route. When I did HS everyone almost without exception did calculus - even those in the arts stream. Now its a lot different and IMHO its a worry.

Thanks
Bill

 

[Dr. Courtney]

In the US there is a tension between producing more scientists, engineers, and medical professions on the one hand and providing a better general math education to all the other professions.

All scientists, engineers, and medical professionals (at the dr level) need calculus and physics. The inability to pass calculus and physics is a huge bottleneck in the number of graduates who can move on into these fields. The lack of STEM grads is more likely to have an economic impact than the suboptimal abilities of other folks in stats.

While it may make sense to throw stats at all the other majors, STEM majors (including pre-med, pre-vet, pre-pharm, and pre-dental) really do need calculus.

And in the physical sciences and engineering, they really need calc-based stats, which means they need calc first.