Thursday, March 13, 2008

Cameras and math skills

Here’s a quote from a popular photography handbook from 1944. This is a book for amateur photographers, not professionals or engineers.

How much longer do you have to make the exposure if you close the aperture from f3.5 to f4.5? Divide the square of 4.5 by the square of 3.5 and you will find with the smaller aperture you must expose 1.67 times as long. Always keep this rule in mind.

That was written at a time when every high school graduate learned enough math to square and divide numbers. We like to think today’s kids have that skill, too, but when I showed that passage to several people here, more than half did not know how to calculate the answer.

Now, look in today’s photography handbooks. You won’t find any calculations. Instead, you find rules of thumb. For example, sixty years later, a 2004 manuals says:

Each click of the aperture ring doubles the exposure. If you need an exposure of 1/60 at f2, you’ll need 1/30 at the next stop, which is f2.8.

By the old formula, we calculate that the exposure must be 1.96 times longer. Using today’s rule, we double it. Both get us essentially the same answer, provided the conditions stay the same. That is, if every lens is marked 2.0, then 2.8. But some lenses are different. What if the lens is marked in fractional stops; 2.0, 2.2, 2.4?

In that scenario, the guy with the formula would get a correct exposure but the guy with a rule would be lost. And that is the value of math. If you know how to calculate the answer, you will never be lost. So why do we ridicule people with math skill? Why is math a “geek class” in high school?

Hobbyist books from 50 years ago are full of math. Hobbyist books today are devoid of math. Why? The need to perform calculation has not gone away. But it has been automated. Fifty years ago, a photographer had to understand exposure calculations because he had to set the camera’s aperture and shutter. Today’s cameras do that for us. And for most people, the result is better exposed pictures. Electronics has replaced math intelligence 99% of the time, for 99% of the people. One insidious result is less capable people.

I'm sure you have experienced the cashier who stands there, frozen and terrified, because she rang up your purchase, then punched $10 for payment, but you handed her a $20. * * * Look of Terror * * * What is the correct change??

We’re on a slippery slope here, letting machines do our calculating for us.

The lack of math skills, and the popular contempt for kids who love math, is dangerous.

17 comments:

Polly Kahl said...

I'm one of those people who isn't gifted in math and needs an automatic camera, but I totally agree with your last three lines.

Daniel J. Watkins said...

Great post John. We need more reminders like this! :)

Jerry Waxler said...

Hi John, the key word is "re" as in "we're on the slope" and already sliding down pretty far. When I was a boy working in my dad's drugstore, I had to make all the change in my head. I loved counting, and perhaps that's the reason I got A's in my calculus classes. Nowadays, (40+ years later, ouch) when I need to calculate a difference, I blank out too. I go inside my head but now I think in words instead of numbers. :)

Jerry
Memory Writers Network

Kim Stagliano said...

Ah, John. We can always COUNT on you for an insightful post. Today, math is taught as a language art - word problem after word problem for the kids. I hate it. My girls excel at memorization - but the verbiage of word problems confuses them. One addition problem says, "How many all together?" The next says "How many do they have?" The next says, "How many does the group have to eat now?" It's terrible and I think discriminatory against boys overall and those with math skill but perhaps not verbal skills as well.

Tammy C said...

I am bad with the hard math but am able to do the mulitplication facts real fast!!

We are showing our age when we talk about using cash registers that made us think on how to count back the money to people-your total was 8.52 and you handed me a 20,okay here is 3 cents to 8.55 2 dimes to 8.74,25 cents to 9.00, a one dollar bill to 10.00 and a 10.00 to 20.00 dollara unless you are short on 10's then it is 2 -5's at 15.00 then 20.00.

anonymoose said...

Not many people butcher their own meat these days either. Most people don't even change their own oil.

Math, division anyway, is a mechanical skill. These days we are focused more on the higher-order functions. There are drawbacks to this of course, but also advantages.

Some folks enjoy the mechanics of the process for the beauty of the process itself, most people just want a nice-enough photo of the family.

One of the triumphs, well the acid-test really, of technology, is how seamlessly it is integrated into the non-technical world.

John is evangelical in his pro-autism stance, an understandable reaction, but I really look forward to the day that we are able to accept it as just another form of 'normal' in the grand distrubution of things...

anonymoose said...

Not many people butcher their own meat these days either. Most people don't even change their own oil.

Math, division anyway, is a mechanical skill. These days we are focused more on the higher-order functions. There are drawbacks to this of course, but also advantages.

Some folks enjoy the mechanics of the process for the beauty of the process itself, most people just want a nice-enough photo of the family.

One of the triumphs, well the acid-test really, of technology, is how seamlessly it is integrated into the non-technical world.

John is evangelical in his pro-autism stance, an understandable reaction, but I really look forward to the day that we are able to accept it as just another form of 'normal' in the grand distrubution of things...

John Elder Robison said...

Anonymoose, I don't think you can equate doing math with butchering meat. You say, "today we are focused on higher order functions."

Tell me what higher order mathematical function we are focused on? I see a diminishment of mathematical capability in society, and that's not just my opinion. Test scores nationwide support it.

You comment on my "pro autism stance," which I have no argument with, but this is not an autism issue. It's a societal problem.

Jan said...

You strike a strong nerve in my heart. The shear beauty of numbers and the relationships of art and architecture, the order of fibonacci sequences in nature and recognition of common mathematical patterns - these are all getting lost today. Only those that study higher mathematics are introduced to these relationships.

One of the beauties of Mathematics is that it is based on logic, concepts and reason. The memorization of formulas, even those for photography, is actually not necessary if one understands simple concepts behind them. The logarithmic relationship of aperture numbers and relationships to photography underly the calculations you discuss. but who knows about Logs? With Slide rule calculations the relationships were more visual. The visual interpretation of time viewing an analogue clock is much more apparent than time passing on a digital. Understanding concepts and the basics of relationships of numbers can be exciting and mind expanding - as well as simplifying the understanding of the order of nature. With a strong basis of these concepts, the rest logically follows.

The use of mathematics, although lessoning in the common world, have gained greater respect in the scientific and even business world than it had 50 years ago. The yin and yang of it all. Because calculations can be done mechanically, it has allowed conceptual mathematicians and statisticians to recognize patterns of numbers more readily. these relationships of digressions, regressions and the like have led to better business and marketing forecasts. If we can maintain a respect for the basic concepts, we will gain a greater understanding of the relationships of spaces, sequences and patterns as our ability to gather and digest more data has evidence patterns we had been missing.

By the way - I read in USA Today

"Math worle celebrates circle of life - Today is a monster day in the world of mathematics; Pi Day. Pi, which represents the ratio of a circle's circumference to its diameter, is celbrated on 3/14 at 1:59:26 p.m. Pi, in case you forgot, equals 3.1415926(and really small change). Today is also Albert Einstein's birthday"

Again, John, you are very timely as well as accurate in you posts!

Thanks

anonymoose said...

John,

This is no doubt true, the diminishment in mathematical function in our society these days. I used the analogy of food preparation, in that a generation or two, or three, ago nearly everybody was intimatly aware of where their meat came from. These days that knowledge is not as essential for most people in day-to-day life as it was. The butchering still gets done, but it's almost unseen in the day to day workings of things. Math is similar. We have other methods these days for achiving the same end-goal. Most people do math for that end-goal, so the easiest, or most efficient means to that end, is usually preferable.

So the higher-order function isn't in the 'doing' of the math, it's in the results. Just like the butchering of the meat.

The higher-order here is in the broad and convenient application of math; calculators, spreadsheets, and statistical programs for example allow us to do with relative ease, what used to take years of specialized training.

Examples like drafting and typing or typesetting as mechanical skills also come to mind. These days the mechanical aspects of these skills are less and less relavent, while their application and use is becoming easier and more-common.

A large number of people, lets call them 'Them' for the sake of the discussion, just want results: they want them quick and easy and simple. They want to know what time it is. Period. Not how a clock works, or the history of time-keeping, or how many time-zones there are, or who it was that invented standard time anyway, and why it was important only after trains started travelling great distances in short time periods...but i digress, but that's because i'm one of 'Us'. 'They' just want to know what time it is. Usually so they'll be on-time for stuff.

Now 'Us', we like the mechanics of things, we like well oiled machines, we oil machines so they run 'better' and build machines just so we can oil them and admire how beautifully they run when well oiled. They don't have to do much to make us happy (except perhaps to be receptive to a good and loving oiling every once and while). 'We' enjoy a good math problem, just, well because. I once worked out pi to 100 decimal places for exactly that reason. Most days though now I'm too lazy to even type in 3.14 and just hit the pi key on the casio.

When You, or I, look at things, we tend to have a very elaborate perspective on the workings, the pulleys and gears, the connections and relations between the parts. We see the beauty and complexity of this and revel in it. We want to know how it works, why it works, what else it can do -- you know the drill. We take things apart, sometimes without thinking about it...

This is one of the beauties of the A.spectrum perspective. We give part-marks. If the answer is wrong, but the solution elegant, we give merit, others just see a wrong answer.

There are benefits to both perspectives of course. One of the blind-spots though is that 'We' don't see that others not only don't see, aren't able to appreciate, the inner-workings. They aren't in-tune with the slick mechanics of just how smoothly the gears meshed -- but they don't care at all about the mechanics of it -- only that they were able to use it to an end.

If we make that end easier 'They' are happy, if 'We' insist that 'They' see the beauty there that we do by explaining in elaborate detail, well -- we know how that goes...

So I think that part of your perception of the 'problem' is that you experience the world in a more mechanically elaborate way that others and thus put more emphasis on this aspect.

That being said, I think everyone should do a little math, butcher an animal or two, or at least hoe a few rows of beans for their dinner and just once, take something apart to see how it really works just for the heck of it...

Kanani said...

I think with the old way, you understood the concept of WHY you have to adjust the aperture. The new way is you're just dialing numbers. Anyway, my son is starting a job at Disneyland. They're teaching him how to count change back the old way. Revelation!

Chris Walrath said...

What we have here is a failure to communicate, or at least to get it. First of all, this is about photography. Or should be. Pure and simple. And photography is math, like it or not. Go to APUG.org and find a legion of photographers that realize you multiply the area of one aperture by the square root of 2 (approx. 1.414) to reach the area of the aperture that will double exposure. And when we go to Wendy's, my 14 year old autistic son has to tell half of the teenagers working there what change to give us. THey have a cash register and they screw it up more oft than not.
Secondly, 'higher functioning math'? Who are we becoming, Stephen freaking Hawking? Gimme a break.
And third, me son is normal to me. Normalcy is a relative concept. Normal things to a normal person are normal. Weird things to a normal person are weird. Weird things to a weird person are normal. And normal things to a weird person are weird.

You do the math.

headless lucy said...

You need to take a light-meter reading as well. This can affect the exposure, too. All cameras work a little differently -- even the same make and brand.

As a general rule, if you open the aperture, increase the shutter speed, and do the opposite if you go the other way.

I've got a camera now that is so advanced, I don't even need it!

Kudos to Steven Wright for that joke.

Chumplet said...

I've always been a fan of bracketing exposure, myself. Take one picture with the aperture open one stop, then the recommended one, and then one below.

Nowadays with digital photography there is no waste. If the shot looks crappy, you delete it.

However, I love the subtle tonal gradations in film.

Math is a sticky wicket for sure. Even fifty years ago, kids argued that they'd never have to use math in real life. They still don't realize how much they need it.

Wendy said...

i was wondering... do you still have obssessions? or are you still interested in one particular subject like when you were little?

John Elder Robison said...

Wendy, when we get older the obsessions often become careers

Jan said...

John - your last comment is quotable! Maybe a t-shirt!

A friend brought me a page from March 24th issue of Forbes... here's a few more quotes of importance and interest...

"The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence is to be found in Mathematics as surely as in peotry"- Bertrand Russell

"To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature" - Richard Feyman


"Mathematics is thought moving in the spere of complete abstraction from any particular instnce of what it is talking about" - Alfred North Whiteheaad

and then from Mae West - "One figure can sometimes add up to a lot.

Finally, Albert Einstein is quoted ass saying "I don't believe in mathematics."


Have a great trip to Boulder!