# You need alegebra and statistics to understand the world

Andrew Sullivan writes a post titled Math is Useless, then follows up with reader reactions. I’m definitely on the side of this guy:

I suppose for a moment that mathematics is only useful “if you engineer planes” or something. When is the decision of whether a person will engineer planes to be made? Should I tell my kids who are in middle school, “Decide now whether you would like to engineer planes, this stuff is useless otherwise.” Should this be decided in high school? Before college? By whom?

It should be noted that without a certain level of mathematical proficiency, about half of the available college majors are essentially eliminated for that student. Many Americans seem to be okay with that choice; students from many other countries are not. Even if one does not wish to spend their lives “engineering planes” an educated populace requires a certain level of numeracy. People rightfully worry about illiteracy; what about innumeracy? Should people be able to solve a simple equation? Understand the notion of compound interest? Percentages? Understand the exploding amount of “data” and “studies” being fed to them daily, especially on unscrupulous “news” sources?

I’d go further than this, though. You need algebra and statistics to understand the world. Without algebra, you won’t be able to understand physics. You won’t be able to understand Newton’s law of gravity. You won’t understand what the hell F = G * m_1 * m_2 / r ^ 2 means. You’ll think Newton is famous because he was the first person to notice that apples fall. And physics may or may not be all  there is to the world, but it does seem to underlie all the parts of the world we even halfway understand. (The sad thing is, if you don’t understand algebra, you have little chance of even understanding the previous sentence.)

The value of statistics should be even more obvious. Medicine, nutrition science, psychology, and sociology all run on statistics. And who doesn’t care about their own health, or about understanding themselves and others? It’s a shame statistics isn’t a required part of high school curricula. (Maybe we could scrap some of the geometry to make room for it? Math-haters do have a point about much of geometry.)

I have this sudden urge to write a book titled The Mathematical Principles Of Knowing What the Fuck Is Going On. Hmm… would anyone be able to translate that into Latin for me?

• unbound

I agree 100%. All aspects of modern life require a functional knowledge of algebra. I personally think statistics should be at least a required course in high school (actually, I think there should be 2 classes on the subject to accommodate probabilities better too).

There are so many things being pushed at us today that use some twist of statistics that people need a much better knowledge of the subject. A lot of the pseudo-science uses statistics improperly (or arrive at bad conclusions). Heck, even a few of the accepted scientific studies use improper conclusions of statistics (evil eye on many health studies). But people are at the mercy of the claims because they have no clue on how to be skeptical about such things.

• Brownian

But people are at the mercy of the claims because they have no clue on how to be skeptical about such things.

That’s just it. Further, people who are mathematically literate—numerate—who have a feel for numbers, can tell when they’re likely to be crap. When I first started my current job (see my comment below), my manager and mentor could immediately look at the results I’d produced and tell that I’d made a mistake in my analysis (and often, exactly what that mistake was) just by what factor my results were off by. He didn’t have to check my numbers against some standard—he could just see that something was fishy, and he’d have me check against a standard. Now, when I train analysts, I try to get them to have this same feel for the data, this intuition (which is really just sensible experience). Without this sense of what the numbers mean, even when they don’t screw up the analysis somewhere along the way they end up producing a lot of meaningless garbage.

• geocatherder

I struggled through this for all of high school, college, and the first couple of years of work (as a computer engineer, yet!) And then one day, fiddling around with some calculation, the gears clicked and I GOT it. I call it the “dance of the numbers” and one either sees it or one doesn’t. If you see it, you can quickly compute in your head the order of magnitude of what an answer should be, if not the exact value. If you understand it, you can easily sort out which real problems are a) algebra problems, aka easy; b) calculus problems, aka difficult in the real world; or c) something more complicated; the cheapest, fastest way to resolve it is to simulate it on a computer and see how it behaves.

Once I sorted all that out, technical life got much easier.

• Brownian

The value of statistics should be even more obvious. Medicine, nutrition science, psychology, and sociology all run on statistics. And who doesn’t care about their own health, or about understanding themselves and others? It’s a shame statistics isn’t a required part of high school curricula. (Maybe we could scrap some of the geometry to make room for it? Math-haters do have a point about much of geometry.)

In his extremely short TED talk, Arthur Benjamin makes the case for retooling high school math education to focus on teaching the skills required for statistical and risk analysis rather than calculus.
I don’t disagree in the least, though I should note that they cover statistics in high school here in Alberta, if only cursorily. The thing about teaching statistics is that the concepts are relatively difficult at first, while the math is relatively easy (I’m employed as a biostatistical analyst*, and I consider myself only passably mathematically literate). But, for most people on a daily basis, statistics is the most useful branch of mathematics besides basic arithmetic.

*Still not quite sure how I faked my way into this gig, other than such subterfuge is the story of my life.

• http://ogremk5.wordpress.com ogremk5

It can be made even more direct to everyone. Almost every working person in the US has a bank account. Without basic algebra skills, you wouldn’t even be able to figure out if the bank is ripping you off or not. Are you just going to trust the bank to be 100% accurate and always play fairly with your money?

If so, then you have bigger issues than not knowing some math.

Many people buy homes at some point in their life, and more buy a car. very few can pay cash, so they must have a loan, compounded annually, on the vehicle or house. Without some degree of sophistication in math, you probably can’t figure out the difference between ‘compounded annually’ and ‘compounded weekly’. That results in a big difference in the amount of money you pay over the life of the loan. Are you going to trust your bank, lender, and car dealership to play fairly with your money and always be 100% accurate?

As an aside, I was absolutely stunned when I read my first contract for a payday loan. A student was using that document as proof of residency and had been doing so for a long time, at least over a year. She was paying 300%APR on those payday loans. If over just a few days, she was losing something like \$50 a month. Just giving it away, so she could have her paycheck a few days early.

Further examples abound when we’re talking about money. Something that everyone has to deal with almost daily. Retirement, stocks, bonds, IRAs, CDs, money markets, loans, heck, even percent off coupons. There is so much potential for a little over top skimming of the unwary. Sure the difference between 4% and 4.01% is tiny, but when magnified among a couple hundred thousand people, the result can be staggering… and people with no math skills will never even realize it.

Oh, and geometry… much of the construction industry depends on it, even if the workers don’t realize it. A friend of mine saved a project because he knew how to bisect an angle, instead of waiting for an engineer to drive 100 miles to do it for the crew.

• Kelly

When I finally admitted to the retired physics professor who runs the local observatory that I’d never taken physics, he said “You’re missing out.” I vaguely sensed that he was right, and now that I’m taking an astronomy class for non-physics majors, I KNOW that he’s right. At 44, I’m undertaking a personal course of remedial math and basic physics, and not only in order to pass the class–I’m gaining satisfaction from learning how the world works.

• Trebuchet

It took me a while to find the followup with user comments on Sullivan’s blog, here’s a liink:

http://andrewsullivan.thedailybeast.com/2012/03/math-is-useless-ctd.html#more

As it happens, before my retirement I did, in fact, engineer planes for a living. A very good living. I struggled with math all the way through high school and college, but with the encouragement of my parents I persevered. I’m glad I did.

• Brownian

BTW, you might be interested to know that WordPress supports LaTeX.

To write Newton’s law of universal gravitation, use the following code without the square brackets around the dollar signs:

[\$]latex F=Gdfrac{m_{1}m_{2}}{r^2}[\$]

becomes

\$latex F=Gdfrac{m_{1}m_{2}}{r^2}\$

Squeeee!

• http://www.loujost.com Lou Jost

Brilliant advice! I had no idea WordPress supported LATEX. I am setting up a quantitative blog and this will change everything.

• machintelligence

I am in agreement with this, but why stop with mathematics education? When most people talk about education reform, they mean doing the same old things in a “better” way. Better usually meaning pouring more facts into kids’ heads and hoping more will stick. I totally agree that statistics is far more useful to most people than geometry, so why not teach it after algebra. The calculus makes physics concepts more intuitive (or at least clearer) so why not introduce it there? It is now pretty well understood that if you want true fluency in a foreign language, you must learn it before age 10. So why are foreign languages only taught in High School? Biology is often the first science class because it is “easier” than Physics or Chemistry, yet real understanding of the important concepts requires knowledge of both those subjects. How about an advanced Biology course for seniors with a strong emphasis on evolutionary theory and biochemistry? Reform should be a total retooling of the curriculum.

• Tsu Dho Nimh

Geometry? If I want to calculate the number of square feet of oak flooring need to re-do a room, I NEED geometry!

• smrnda

As a person with advanced degrees in mathematics and computer science, if people knew math better, we would have a very different political landscape.

I recall one person who just couldn’t believe that going to a flat tax could both raise taxes on most people (this was the Herman Cain 9 9 9 plan we were discussing) while still causing a decrease in tax revenue. Anybody with a sixth grade understanding of pre-algebra concepts like percents, multiplication and addition, would see through.

• smrnda

On statistics and calculus though, you do need calculus for any serious sort of statistics.

• machintelligence

I disagree. Calculus might make the derivation of some statistical tests easier, but you certainly don’t need it to apply them. Once you understand the basis for inferential statistics (not that some multivariate statistics with repeated measures aren’t complicated) and have the right program package, you are good to go.

• mnb0

“Without algebra, you won’t be able to understand physics.”
Fill in about every other branch of science and you have the point. Mathematics is not useful in itself, but it is the single most important subject as an application to other fields. Remember: as soon as a child learns to count to ten it learns maths.

• iknklast

You also need algebra and statistics to understand Ecology, which is more important than most people recognize. Most people at the college where I teach think Ecology is about going out and looking at plants and animals in the wild; I’ve never been able to convince them that the real work of Ecology is done later, when you process that data through all sorts of multivariate statistics.

My mother used to claim she’d never used math after she got out of high school, but I watched her in the grocery store every week calculate the value of different products that had different weights, and figure out which was the best value for the money. My husband claims he can’t do statistics, but he calculates baseball stats all the time.

Math is life. If we have no understanding of math, it’s easy to be taken to the cleaners, like the people who go to Sam’s club and think the big case they’re buying is always a good value, because they don’t know how to figure the price of an individual item and realize that some of the things they could get at the local grocery store cheaper.

• machintelligence

As a former practicing ecologist, I couldn’t agree more. When I was in graduate school (40 years ago!) I was the statistics maven for all of the other grad students. Some of the faculty weren’t too statistically literate either — but they knew how to do sums of squares on a mechanical calculator. Biologists who could program were rare in those days.
Your parents were probably of the opinion that arithmetic wasn’t math and descriptive statistics weren’t real statistics. There is a shred of truth in this, but they were obviously not innumerate.
Another pet peeve: Have you ever gone shopping for a car and been told when asking for the price “it will be \$408 per month”? I remember one instance where we finally got an “out the door” price (total dollars) and then had to fight with the finance dept. because they had added all sorts of extras (extended warranty, paint protector, etc.) Fortunately I had brought a financial calculator along and calculated the correct monthly payments.

• piero

As a mathematician myself, I admit I’m slightly biased. But I could not agree more with Chris’s post.

For thos who want to pursue this issue further, I’d recommend “Innumeracy: Mathematical Illiteracy and Its Consequences”, bu John Allen Paulos (http://www.amazon.com/Innumeracy-Mathematical-Illiteracy-Consequences-Vintage/dp/0679726012)

A simple example: I’ve come to realize that most people think that a diminishing inflation rate means lower prices, whereas in fact it just means that prices are rising more slowly.

I the price of a refrigerator decreases by 5% and then increases by 5%, is the final price the same as the original one? Most students will answer that it will obviously be the same, thus proving that “obviously” is a word to be seldom used.

Another example: millions of people buy lottery tickets, and chances are they will spend a fair amount of money and never get a prize in their whole lives. It is an absolutely irrational behaviour, yet…

• jacobfromlost

Innumeracy often maximizes profits also. From my time in retail, I can’t say I ever met a customer that understood percentages, and that ignorance was ROUTINELY used to trick people into buying things they probably would not have bought if they understood what the final price would be.

Years ago there was a Candid Camera episode where there was a display of expensive shoes or something, and the sign said, “112% off”. They had dozens of people who tried to calculate how much that would be, and the typical answer was “a few dollars”.

Of course, in a country that is financially illiterate, it isn’t surprising that they find math to be useless.

• iknklast

I had a student who answered on a test that the world was 100% covered with water, without understanding what that meant – since the paper they handed in wasn’t soggy with the ink running everywhere, I felt compelled to count it wrong, with commentary.

I don’t think they read the commentary, because on the next test, the same student indicated that 100% of all species that had ever lived are now extinct. Ouch.

The worst of it? Many of my students are making As in Algebra (and some even in Calculus) while they can’t calculate a simple percentage, don’t understand order of operations, and can’t figure out a ratio to save their lives. In fact, I had a student who made an A in Calculus, and the entire semester in my Physical Geography class, she didn’t do a single calculation correctly. On tests, she usually skipped calculations. And the math in that class is much simpler than in Calculus!

My associate dean explained it all, however; I just wasn’t teaching to the correct “learning style” – psychobabble which appears to have little or no meaning in the real world, except as a blunt weapon to bludgeon teachers over the head with, so the students have an excuse that doesn’t involve “Maybe I should hve studied more”.

• SAWells

I had a student put on their exam that “Outside the planet there is no mass and so the gravitational field is zero”. Since they weren’t levitating at the time – I know, I invigilated the exam – I had to knock some marks off…

• jacobfromlost

iknklast: I just wasn’t teaching to the correct “learning style” – psychobabble which appears to have little or no meaning in the real world, except as a blunt weapon to bludgeon teachers over the head with, so the students have an excuse that doesn’t involve “Maybe I should hve studied more”.

Me: I was a high school English teacher, so I know where you are coming from. I do think different people have different learning styles, and it is helpful and useful to keep that in mind when lesson planning, but more often than not it is used as an excuse to beat teachers over the head if something goes wrong. As soon as the kids realize that the teacher will be blamed if they do poorly…what motivation do they have to do well? Especially at skills that may be difficult at first? Teachers are put in a lose-lose situation daily.

(I came to realize after 8 years that those in charge had absolutely no idea what they were doing–they just repeat things like the “learning styles” thing and feel their job is done. I once had a class from hell where a particularly troubled girl was transferred from 100 miles away after starting too many fights at her old school. The girl later got into a fight at lunch, and the administrator tried to blame ME for that! She told me, “I’m not sure if her fighting was from being in your class. I just don’t know.” What? How do you defend yourself against that? And how to you explain that even HINTING at this kind of attitude to the kids gives them license to act however they wish, and blame any teacher they like–even if that teacher isn’t even around at the time. I actually knew a teacher who got marked down on only one thing–that the kids didn’t act very well when he was absent!)

• http://rockstarramblings.blogspot.com/ Bronze Dog

I remember when I was in middle school trying to explain to a friend that “over 100% off” would mean a negative price, but he kept saying, “They did it with calculators!” If we had been in class, that would have been an appropriate time to headdesk.

I should clarify my comment about geometry: my impression is that geometry, as done by the ancient Greeks, has been made largely obsolete by the Cartesian coordinates (a.k.a. doing geometry with algebra), mass-produced precision measuring tools, and computers that can do trigonometry.

So ogremk5, when you say there was an issue with knowing how to bisect an angle, what do you mean? Do you just mean they didn’t know what the word “bisect” meant? Or was there some reason why they needed to do it via compass and straitedge or something, rather than just measuring the angles?

And Tsu Dho Nimh, I’m talking the more esoteric proofs and methods of geometry, not the basic concepts like surface area and volume that can be covered in algebra or even pre-algebra.

I have just started handing over this book when I get told that only engineers or mathematicians use math.

I point out some of the food calculations and they look pretty sheepish.

• http://becomingjulie.blogspot.com/ BecomingJulie

One thing I’d change in the teaching of maths is to do away with mixed integers and ratiometric fractions (e.g. 3¾, as opposed to 15/4 or 3.75) altogether. They are as obsolete as Roman numerals.

After all, everything in The Real World — weighing scales, tape measures, measuring jugs, money, voltmeters — is decimal. Any other way of expressing fractions is just adding needless confusion.

• daved

The tape measure in my toolbox is marked off in 1/16ths, as I recall, and the scales in the produce department of my local grocery store work in pounds and ounces (and halves or quarters of ounces).

Also, many common fractions do not have exact decimal representations. Why don’t you go out and write out the exact decimal representation of 1/3 for me? I’ll wait.

• http://becomingjulie.blogspot.com/ BecomingJulie

You really need to get yourself a new tape measure, then! Mine is marked in millimetres, of which there are exactly 1000 in a metre. The weighing scales in local shops are also subdivided by thousands (grams in a kilogram this time). You can even verify that sale price = mass * specific price with your own calculator.

Your second request is also a bit of a straw dummy. Nobody ever needs to write the final decimal representation of 1/3 (or any other fraction) with any more precision than the instrument they are ultimately going to be measuring it with. Grocers’ scales usually read to the nearest .005 kg., so “a third” of a kilo of stuff on them would be .330 or .335 kg. Kitchen measuring jugs only read to the nearest 25 ml. or so, so 1/3 of a litre would be 325 ml. If I needed more precision, I’d make it up to 332.5 ml. using a 2.5 ml. measuring spoon.

Of course I’d leave the decimal fraction on the calculator at full precision while e.g. working out positions of shelves, mentally rounding to 3 places each time.

• http://becomingjulie.blogspot.com/ BecomingJulie

Let’s try a real-world example: fixing 6 shelves in an alcove 2.28 m. high, evenly spaced and with the bottom shelf 1 m. above floor level. I’ll be using a tape measure marked in metres, centimetres and millimetres, and an ordinary 8-digit calculator from a pound store. Key presses are in bold, figures displayed on the calculator are in italics.

2.28 [-] 1 [=] (get the height of the shelves plus the space above top shelf)
1.28
[÷] 6 [=] (divide this by the number of shelves)
0.2133333 (the spacing between each shelf and the next one)
[+] [+] (make this a constant for addition. Now, each subsequent press of the [=] key will add 0.2133333 to whatever is on the calculator’s display).
1 [=] (we marked the position of the bottom shelf at 1m. Now we want to start by adding 0.2133333 to 1, to get the position of the second shelf up.)
1.2133333 (mentally round this to 1.213 — the same precision as the tape measure — and mark the position of the second shelf on the wall)
[=]
1.4166667 (mentally round this to 1.417 and mark 3rd shelf. Note in passing that this calculator seems to be using more precision than it can display. This is not a bad thing.)
[=]
1.64 (mark 4th shelf)
[=]
1.8533333 (mark 5th shelf)
[=]
2.0666667 (mark last shelf)
[=] (one final time just to check; if all has gone to plan, we should get the height of the ceiling)
2.28 (Yay! Already time for a brew and a smoke, while you’re still fart-arsing about trying to subtract 3 ft. 3 3/8 in. from 7 ft 5 3/4 in. Hey, what was that noise? It sounded a bit like a space probe crashing …..)

• mnb0

I (teacher maths and physics) teach my kids to rewrite 3 3/4 immediately as 15/4 and only to work 15/4 out at the very end of their calculation. No, I would not mind at all if mixed integers were abolished.

• http://ogremk5.wordpress.com ogremk5

Chris@15

This was a long time ago, but the problem was that they had to cut a pipe or something and they needed to bisect the angle for the cut. As a rule construction workers don’t have protractors and compasses lying around.

So my buddy was able to draw the line to cut on using a pencil and some string.

A few years ago I was talking to my dad about the need for everyone to know calculus. He countered that he had managed perfectly well for six decades, two mortgages, half a dozen cars and innumerable budgets without advanced math. We have had this talk a number of times to no avail, but suddenly something clicked. “But dad, when you got your first mortgage, nobody offered you an adjustable rate. Things have changed.” That would have been October ’08 and for the first time, it seemed to register.

Calculus rocked my world more than any other class I have ever taken. Understanding derivatives and integrals fundamentally changed the way I saw everything. I started driving differently. I started spending differently. Nothing was unaffected. Sure, you can go through life and never use math. When a kid asks ‘when are we ever going to use this in real life?’, the answer is probably never. But it will be a life disempowered from what could have been and vulnerable to simple but catastrophic mistakes.

• http://ogremk5.wordpress.com ogremk5

Another one that just happened to me this weekend.

30% a price that is already 60% off does not mean that the current price is 90% off from the original price.

I had to explain that one to my mom… sigh.

• piero

Sadly, most people fall for that one. One day the shop advertises a 40% off. Two days later, it adevertises a further 40% off. Te number of people who will realize that they’ll pay far more than 20% of the original price can be counted on the fingers of a fingerless hand.

• Caravelle

Surprise, another voice here from a maths-understanding person saying it’s really important to understand maths. But really I think it isn’t just the issue of fighting against those who say you don’t need maths; it’s a matter of teaching maths right.

For one thing, maths isn’t about following blind recipes; if that’s all you do it can be better than nothing I guess, but in a way the word “blind” there is very appropriate. You need a sense of how mathematics work, of why the right answer is right and what it should roughly be.

And I think that maths sense can be taught (… or can be induced into children through various teaching methods if you prefer). It just really isn’t, not systemically.

But I suppose that’s a bit ambitious in that pedagogy is still an art to a great extent. Failing that, have every schoolchild read How to lie with statistics. That would be a start.

• piero

Indeed. And I would also recommend “The psychology of learning matehematics” by Richard Skemp, any book by Martin Gardner and the evergreen classic “What is mathematics” by Courant and Robbins.

• wholething

Many years ago, I learned to find the cube root of a perfect cube of a two digit number by inspection and wanted to try it out at work. I handed people a calculator and asked them to cube any two digit number. Everybody understood what a two digit number was but less than half knew what the cube a number was.

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